Introduction to Inequalities

An inequality is a mathematical statement that compares two quantities that are not necessarily equal.

An equation tells us that two expressions have exactly the same value:

x = 7

An inequality tells us that one quantity may be greater than, less than, greater than or equal to, or less than or equal to another quantity:

x > 7

This means that x can be any number greater than 7, such as 8, 10, 25, or 7.5.

Unlike many equations, inequalities usually have many possible solutions rather than just one.

Equations and Inequalities

An equation uses an equals sign:

=

For example:

x + 3 = 8

There is one value of x that makes this equation true:

x = 5

An inequality uses an inequality symbol:

<, >, ≤, or ≥

For example:

x + 3 < 8

Many values make this statement true:

x = 4

x = 2

x = 0

x = −10

All of these values satisfy the inequality.

 

 

 

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The Main Inequality Symbols

There are four main inequality symbols.

The difference between < and ≤ or between > and ≥ is extremely important.

Less Than

The symbol:

<

means less than.

For example:

x < 6

means that x can be any number smaller than 6.

Possible solutions include:

• 5
• 3
• 0
• −4
• 5.9

However:

x = 6

is not a solution because the inequality does not include 6.

Greater Than

The symbol:

>

means greater than.

For example:

x > 4

Possible solutions include:

• 5
• 6
• 10
• 4.1
• 100

But:

x = 4

is not a solution.

Less Than or Equal To

The symbol:

≤

means less than or equal to.

For example:

x ≤ 7

means:

x < 7 or x = 7

Possible solutions include:

• 7
• 6
• 0
• −10
• 6.99

Because the inequality contains or equal to, 7 itself is included.

Greater Than or Equal To

The symbol:

≥

means greater than or equal to.

For example:

x ≥ 3

means:

x > 3 or x = 3

Possible solutions include:

• 3
• 4
• 10
• 3.5
• 1000

The value 3 is included.

Reading Inequalities Correctly

The direction of the inequality symbol matters.

Compare:

x < 8

and:

x > 8

The first means:

x is less than 8

The second means:

x is greater than 8

A useful way to remember the symbols is that the wide opening points toward the larger number.

For example:

3 < 9

The wider side faces 9 because 9 is larger.

Similarly:

12 > 5

The wider side faces 12 because 12 is larger.

Reading Inequalities from Either Direction

An inequality can be written in either direction.

For example:

x < 6

means exactly the same thing as:

6 > x

Both statements say that x is smaller than 6.

Similarly:

x ≥ 10

means the same thing as:

10 ≤ x

When the order is reversed, the inequality symbol must also reverse direction.

Mathematical Statements from Words

Words can be translated into inequality notation.

A number is greater than 12

becomes:

x > 12

A number is less than 20

becomes:

x < 20

A number is at least 15

becomes:

x ≥ 15

A number is at most 15

becomes:

x ≤ 15

The phrases at least and at most are particularly important.

Understanding "At Least"

At least 10 means 10 or more.

Therefore:

x ≥ 10

For example:

"You must be at least 16 years old."

means:

age ≥ 16

A person who is exactly 16 satisfies the condition.

Understanding "At Most"

At most 10 means 10 or less.

Therefore:

x ≤ 10

For example:

"The elevator can carry at most 12 people."

means:

number of people ≤ 12

Twelve is allowed, but thirteen is not.

Other Common Inequality Phrases

Recognizing these phrases makes it much easier to translate real situations into mathematics.

Solution Sets

The solution set of an inequality contains every value that makes the inequality true.

Consider:

x > 5

The solution set includes:

6, 7, 8, 9, ...

but it also includes:

5.1, 5.01, 5.001, ...

There are infinitely many solutions.

The solution is therefore better described as:

all real numbers greater than 5

rather than trying to list every possible value.

Testing a Possible Solution

To determine whether a number belongs to a solution set, substitute it into the inequality.

Consider:

x < 9

Is x = 6 a solution?

Substitute:

6 < 9

This statement is true.

Therefore:

6 is a solution.

Testing a Value That Does Not Work

Consider:

x > 4

Is x = 2 a solution?

Substitute:

2 > 4

This statement is false.

Therefore:

2 is not a solution.

Testing the Boundary Value

The number that separates solutions from non-solutions is often called the boundary value.

Consider:

x < 5

The boundary is 5.

Test it:

5 < 5

False.

Therefore, 5 is not included.

Now consider:

x ≤ 5

Test the boundary:

5 ≤ 5

True.

Therefore, 5 is included.

This difference determines how the inequality is represented on a number line.

Inequalities on a Number Line

A number line provides a visual representation of an inequality.

Two things must be shown:

• whether the boundary value is included
• which direction contains the solutions

For:

x > 4

the solutions are all numbers to the right of 4.

For:

x < 4

the solutions are all numbers to the left of 4.

 

 

 

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Open Circles

An open circle means that the boundary value is not included.

Use an open circle for:

<

and:

>

For example:

x > 3

Place an open circle at 3 and extend the solution line to the right.

Why an open circle?

Because:

3 > 3

is false.

Closed Circles

A closed circle means that the boundary value is included.

Use a closed circle for:

≤

and:

≥

For example:

x ≥ 3

Place a closed circle at 3 and extend the solution line to the right.

Why?

Because:

3 ≥ 3

is true.

Direction on a Number Line

Numbers increase as we move to the right.

Numbers decrease as we move to the left.

Therefore:

x > 5

extends right.

x ≥ 5

extends right.

x < 5

extends left.

x ≤ 5

extends left.

This is often more reliable than trying to memorize which way an arrow should point.

Example: x < 2

The inequality:

x < 2

means all numbers smaller than 2.

The number-line representation has:

• an open circle at 2
• solutions extending left

Examples of solutions:

1, 0, −1, −10, 1.9

Not solutions:

2, 3, 100

Example: x ≤ 2

Now consider:

x ≤ 2

The number-line representation has:

• a closed circle at 2
• solutions extending left

The difference is that:

x = 2

is now included.

Example: x > −3

Negative numbers sometimes make inequalities seem more difficult.

Consider:

x > −3

Solutions include:

• −2
• −1
• 0
• 5
• 100

Remember that −2 is greater than −3 because it lies farther to the right on the number line.

The graph has:

• an open circle at −3
• solutions extending right

Comparing Negative Numbers

On a number line:

−8 < −3

because −8 is farther left.

Similarly:

−2 > −7

because −2 is farther right.

A useful rule is:

farther right = greater

farther left = smaller

This works for positive numbers, negative numbers, and zero.

Writing an Inequality from a Number Line

Suppose a number line shows:

• a closed circle at 6
• shading extending to the right

The closed circle means:

equal to is included

The rightward direction means:

greater than

Therefore:

x ≥ 6

Another Number-Line Example

Suppose a number line shows:

• an open circle at −4
• shading extending left

Open circle:

not equal to −4

Left:

less than −4

Therefore:

x < −4

From Words to a Number Line

Consider:

A temperature below 5°C

"Below" means:

T < 5

Therefore:

• open circle at 5
• solutions extend left

Now consider:

A temperature of 5°C or below

This means:

T ≤ 5

Therefore:

• closed circle at 5
• solutions extend left

A small change in wording changes whether the boundary is included.

Real-World Inequalities

Inequalities are useful because many real situations involve limits rather than exact values.

Examples include:

• age restrictions
• speed limits
• weight limits
• budgets
• temperatures
• minimum scores
• maximum capacities
• height requirements

 

 

 

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Maximum Capacity

An elevator has a maximum capacity of:

900 kg

Let m represent the total mass inside the elevator.

Then:

m ≤ 900

The elevator may contain exactly 900 kg, so the equality is included.

Minimum Score

A student needs at least:

70%

to meet a particular requirement.

Let s represent the score.

Then:

s ≥ 70

A score of exactly 70 satisfies the requirement.

Speed Limit

Suppose the maximum permitted speed is:

50 km/h

Let v represent speed.

Mathematically, the permitted speeds can be represented as:

v ≤ 50

In a realistic context, speed would also normally satisfy:

v ≥ 0

Context often places additional restrictions on a mathematical variable.

Temperature Example

Water is stored at temperatures below:

5°C

Let T represent temperature.

"Below 5°C" means:

T < 5

If instead the requirement were:

"5°C or below"

then:

T ≤ 5

The wording determines whether equality is allowed.

Age Example

A competition is open to participants who are:

18 years old or older

Let a represent age.

Then:

a ≥ 18

Because someone who is exactly 18 is allowed, the symbol must include equality.

Inequality vs Equation in Context

Suppose a theater has exactly 250 seats.

The statement:

s = 250

describes the number of seats.

But the number of people who can be seated cannot exceed 250:

p ≤ 250

The first statement is an equation.

The second is an inequality.

Equations often describe an exact value, while inequalities frequently describe a range of acceptable values.

Finite and Infinite Solution Sets

Some inequalities have infinitely many possible solutions.

For example:

x > 2

has infinitely many real-number solutions.

However, context can restrict the solution set.

Suppose:

"A classroom can contain at most 30 students."

Then:

s ≤ 30

But the number of students must normally be a whole number and cannot be negative.

The realistic solution set is therefore:

0, 1, 2, 3, ..., 30

Context matters when interpreting inequalities.

Integer Solution Sets

Suppose:

x < 4

and x must be a positive integer.

Then the solution set is:

{1, 2, 3}

If x can be any real number, however, there are infinitely many solutions below 4.

Always check what type of numbers the variable represents.

Writing Solution Sets

For simple inequalities, solution sets can be described in several ways.

For:

x ≥ 4

we could write:

x ≥ 4

or say:

all real numbers greater than or equal to 4

or show the solution on a number line.

Each representation communicates the same solution set.

Boundary Values

The boundary value is especially important when graphing an inequality.

For:

x > 8

boundary = 8

For:

x ≤ −2

boundary = −2

For:

y ≥ 0

boundary = 0

Always ask:

Is the boundary itself a solution?

If yes, use ≤ or ≥ and a closed circle.

If no, use < or > and an open circle.

Comparing Several Inequalities

Consider:

x > 5

and:

x ≥ 5

Both contain numbers greater than 5.

Only the second includes 5.

Now compare:

x < 5

and:

x ≤ 5

Both contain numbers smaller than 5.

Only the second includes 5.

The equality bar underneath < or > changes the solution set by including the boundary value.

Checking Solutions Efficiently

Suppose:

x ≥ −2

Determine whether each value is a solution.

x = −5

−5 ≥ −2 is false.

Not a solution.

x = −2

−2 ≥ −2 is true.

Solution.

x = 0

0 ≥ −2 is true.

Solution.

x = 10

10 ≥ −2 is true.

Solution.

A number line makes this pattern easy to see.

Matching Words and Symbols

Consider each phrase.

More than 20

x > 20

No more than 20

x ≤ 20

At least 20

x ≥ 20

Less than 20

x < 20

The words no more than are sometimes confused with more than, but they mean very different things.

Compound Ideas in Everyday Language

Even before formally studying compound inequalities, some situations naturally involve two limits.

For example:

"A temperature between 10°C and 20°C."

This tells us that the temperature has both a lower and upper boundary.

If both endpoints are included:

10 ≤ T ≤ 20

This means:

T is at least 10 and at most 20.

This idea will become important when working with more advanced inequalities.

Why Number Lines Are Useful

A number line shows the entire solution set at once.

For example, if:

x ≥ 2

we can immediately see that:

• 2 is included
• 3 is included
• 100 is included
• 1 is not included
• −5 is not included

A graph therefore provides information that a single example cannot.

Inequalities and Measurement

Measurements naturally involve inequalities because real measurements often have acceptable ranges.

Suppose a machine part must have a mass no greater than:

2.5 kg

Then:

m ≤ 2.5

Or suppose a plant requires temperatures above:

15°C

Then:

T > 15

Inequalities allow mathematical models to describe constraints and acceptable conditions.

Inequalities and Constraints

A constraint is a restriction placed on possible values.

For example, a suitcase may have a maximum mass of 23 kg.

The constraint is:

m ≤ 23

A roller coaster may require riders to be at least 140 cm tall:

h ≥ 140

A budget may allow spending of no more than $500:

C ≤ 500

Inequalities are therefore fundamental in planning and decision-making.

A Strategy for Interpreting Inequalities

When reading an inequality, follow these steps:

1. Identify the variable.
2. Identify the boundary value.
3. Determine whether values greater or smaller than the boundary are allowed.
4. Determine whether the boundary itself is included.
5. Write the correct inequality symbol.
6. Check the statement using one possible value.
7. Represent it on a number line if needed.

For:

"Tickets cost no more than $25."

Variable:

c = ticket cost

Boundary:

25

"No more than" means 25 or less.

Therefore:

c ≤ 25

Common Mistakes

Confusing < and >

Remember that the wide opening faces the greater value.

Forgetting the equality part of ≤ and ≥

The line underneath means that the boundary value is included.

Thinking x > 5 means x = 6 only

An inequality usually describes many possible values.

Using a closed circle for < or >

Strict inequalities use open circles.

Using an open circle for ≤ or ≥

These symbols include the boundary, so use a closed circle.

Graphing x < 5 to the right

Numbers less than 5 are located to the left of 5.

Thinking −8 is greater than −3

−8 lies farther left, so:

−8 < −3

Misreading "at least"

At least 10 means:

x ≥ 10

not x ≤ 10.

Misreading "at most"

At most 10 means:

x ≤ 10

not x ≥ 10.

Ignoring the context

A mathematical inequality might allow decimals or negative values even when the real situation does not.

Key Terms

Equation — mathematical statement showing that two expressions have equal values.

Inequality — mathematical statement comparing quantities that may not be equal.

Inequality symbol — symbol such as <, >, ≤, or ≥ used to compare quantities.

Less than (<) — indicates that the quantity on the left is smaller.

Greater than (>) — indicates that the quantity on the left is larger.

Less than or equal to (≤) — indicates that a value is smaller than or equal to another value.

Greater than or equal to (≥) — indicates that a value is larger than or equal to another value.

Solution — value that makes a mathematical statement true.

Solution set — collection of all values that satisfy an inequality.

Boundary value — value separating solutions from non-solutions.

Open circle — number-line symbol showing that a boundary value is not included.

Closed circle — number-line symbol showing that a boundary value is included.

Number line — visual representation of numbers arranged from smaller to larger.

Constraint — restriction placed on the possible values of a variable.

Variable — symbol representing an unknown or changing quantity.

Key Takeaways

• An equation states that two quantities are equal.
• An inequality compares quantities that are not necessarily equal.
• The four main inequality symbols are <, >, ≤, and ≥.
• < means less than.
• > means greater than.
• ≤ means less than or equal to.
• ≥ means greater than or equal to.
• Inequalities usually have many possible solutions.
• The solution set contains every value that makes an inequality true.
• A possible solution can be checked by substituting it into the inequality.
• At least means greater than or equal to.
• At most means less than or equal to.
• On a number line, greater values are to the right and smaller values are to the left.
• Use an open circle when the boundary is not included.
• Use a closed circle when the boundary is included.
• The boundary is included for ≤ and ≥ but not for < and >.
• Negative numbers still follow the rule that numbers farther right on the number line are greater.
• Real-world inequalities often represent limits, minimums, maximums, and acceptable ranges.
• Context can restrict a solution set to positive numbers, whole numbers, or other meaningful values.

Check Your Understanding

1. Explain the difference between an equation and an inequality.
2. What does each symbol mean: <, >, ≤, ≥?
3. Write an inequality representing: x is greater than 12.
4. Write an inequality representing: y is less than or equal to 8.
5. Write an inequality representing: a number is at least 20.
6. Write an inequality representing: a number is at most −3.
7. Is x = 6 a solution of x > 4? Explain how you know.
8. Is x = 5 a solution of x < 5? What about x ≤ 5?
9. Describe how you would represent x > −2 on a number line.
10. Describe how you would represent x ≤ 4 on a number line.
11. A number line has a closed circle at −3 and extends to the right. Write the corresponding inequality.
12. A number line has an open circle at 7 and extends to the left. Write the corresponding inequality.
13. A ride requires passengers to be at least 140 cm tall. Define a variable and write an inequality representing the requirement.
14. A container can hold no more than 25 kg. Write an inequality representing its possible mass and explain why the boundary value is included.
15. A student writes x ≤ 10 for the statement "x is at least 10." Explain the error, write the correct inequality, and describe how its number-line representation should look.

Solving One-Step Inequalities

A one-step inequality is an inequality that can be solved using one inverse operation.

For example:

x + 4 > 9

can be solved by subtracting 4 from both sides:

x + 4 − 4 > 9 − 4

x > 5

The solution is not just one number. It represents every value greater than 5.

Solving inequalities is similar to solving equations, but there is one especially important difference:

When multiplying or dividing both sides of an inequality by a negative number, the inequality symbol must reverse direction.

Inequalities and Equations

Consider the equation:

x + 3 = 10

Subtract 3 from both sides:

x = 7

There is one solution.

Now consider:

x + 3 < 10

Subtract 3 from both sides:

x < 7

There are infinitely many solutions.

Examples include:

6, 5, 0, −10, and 6.9.

The goal when solving an inequality is to find the entire solution set.

Inverse Operations

An inverse operation reverses another operation.

To isolate the variable, use the inverse operation on both sides of the inequality.

Solving Addition Inequalities

Consider:

x + 5 > 12

The variable has 5 added to it.

Use the inverse operation and subtract 5 from both sides:

x + 5 − 5 > 12 − 5

Therefore:

x > 7

The solution is every number greater than 7.

Checking the Solution

Choose a number that satisfies:

x > 7

For example:

x = 10

Substitute into the original inequality:

10 + 5 > 12

15 > 12

This is true.

Therefore, 10 is a valid solution.

Representing x > 7 on a Number Line

For:

x > 7

the boundary value is 7.

Because 7 itself is not included, use:

• an open circle at 7
• shading or an arrow extending to the right

The line extends right because numbers greater than 7 are found to the right.

 

 

 

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Another Addition Example

Solve:

x + 8 ≤ 15

Subtract 8 from both sides:

x + 8 − 8 ≤ 15 − 8

x ≤ 7

Because 7 is included, the number line has:

• a closed circle at 7
• shading extending left

Check using x = 7:

7 + 8 ≤ 15

15 ≤ 15

True.

Solving Subtraction Inequalities

Consider:

x − 6 < 4

Add 6 to both sides:

x − 6 + 6 < 4 + 6

x < 10

The number line has:

• an open circle at 10
• shading extending left

Another Subtraction Example

Solve:

x − 9 ≥ 3

Add 9 to both sides:

x − 9 + 9 ≥ 3 + 9

x ≥ 12

Check using the boundary value:

12 − 9 ≥ 3

3 ≥ 3

True.

Therefore, 12 is included in the solution set.

Adding and Subtracting Negative Numbers

Negative numbers do not change the basic method.

Consider:

x + 7 > −2

Subtract 7:

x > −2 − 7

x > −9

Now consider:

x − 4 ≤ −10

Add 4:

x ≤ −10 + 4

x ≤ −6

Pay careful attention to integer arithmetic.

Example with a Negative Constant

Solve:

x + 6 ≥ −3

Subtract 6:

x ≥ −3 − 6

x ≥ −9

Check x = −9:

−9 + 6 ≥ −3

−3 ≥ −3

True.

Solving Multiplication Inequalities

Consider:

3x > 15

The variable is multiplied by 3.

Divide both sides by 3:

3x/3 > 15/3

x > 5

Because we divided by a positive number, the inequality symbol does not change.

Another Multiplication Example

Solve:

4x ≤ 24

Divide both sides by 4:

4x/4 ≤ 24/4

x ≤ 6

The number-line representation has:

• closed circle at 6
• shading extending left

Solving Division Inequalities

Consider:

x/5 > 3

Multiply both sides by 5:

x > 15

Because 5 is positive, the inequality direction remains unchanged.

Another Division Example

Solve:

x/4 ≤ 6

Multiply both sides by 4:

x ≤ 24

Check the boundary:

24/4 ≤ 6

6 ≤ 6

True.

Therefore, 24 is included.

The Special Rule for Negative Numbers

The most important special rule when solving inequalities is:

If you multiply or divide both sides by a negative number, reverse the inequality symbol.

For example:

−2x > 8

Divide both sides by −2.

Because we divide by a negative number:

> becomes <

Therefore:

x < −4

 

 

 

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Why Does the Inequality Reverse?

Start with a true statement:

3 < 7

Multiply both sides by −1:

3(−1) and 7(−1)

giving:

−3 and −7

But:

−3 > −7

not:

−3 < −7

Multiplication by a negative number reflects the numbers across zero on the number line.

Their order reverses.

Therefore:

3 < 7

becomes:

−3 > −7

Visualizing the Negative Rule

On a number line:

2 < 5

because 2 is to the left of 5.

Multiply both numbers by −1:

−2 and −5

Now −2 is to the right of −5.

Therefore:

−2 > −5

The order has reversed.

This is why the inequality symbol must reverse when multiplying or dividing by a negative number.

Solving with Negative Multiplication

Solve:

−3x < 12

Divide both sides by −3.

Because −3 is negative, reverse:

< to >

Therefore:

x > −4

Check with x = 0:

−3(0) < 12

0 < 12

True.

So 0 belongs to the solution set.

Another Negative Multiplication Example

Solve:

−5x ≥ 20

Divide both sides by −5.

Reverse ≥ to ≤:

x ≤ −4

Check x = −4:

−5(−4) ≥ 20

20 ≥ 20

True.

The boundary value is included.

Solving Division by a Negative Number

Consider:

x/−4 > 3

Multiply both sides by −4.

Because we multiply by a negative number, reverse the symbol:

x < −12

Check:

x = −16

Original inequality:

−16/−4 > 3

4 > 3

True.

Therefore, −16 is a solution.

Another Negative Division Example

Solve:

x/−6 ≤ 2

Multiply both sides by −6.

Reverse ≤ to ≥:

x ≥ −12

Check the boundary:

−12/−6 ≤ 2

2 ≤ 2

True.

Therefore, −12 is included.

When Do You Reverse the Symbol?

Reverse the inequality symbol only when multiplying or dividing both sides by a negative number.

Do not reverse it when:

• adding a positive number
• adding a negative number
• subtracting a positive number
• subtracting a negative number
• multiplying by a positive number
• dividing by a positive number

This distinction is extremely important.

Example: Subtracting a Negative Number

Solve:

x + 4 > 9

Subtract 4:

x > 5

The inequality does not reverse.

Now consider:

x − (−4) > 9

Since subtracting −4 is the same as adding 4:

x + 4 > 9

Again:

x > 5

The presence of a negative number alone does not cause the inequality to reverse.

The reversal happens specifically when both sides are multiplied or divided by a negative number.

Four Possible Symbol Reversals

When multiplying or dividing by a negative:

< becomes >

> becomes <

≤ becomes ≥

≥ becomes ≤

For example:

−2x ≤ 10

becomes:

x ≥ −5

Graphing Solutions on a Number Line

Once an inequality has been solved, represent the solution using:

• a boundary point
• an open or closed circle
• a direction

Use an open circle for:

< or >

Use a closed circle for:

≤ or ≥

 

 

 

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Direction of the Graph

For:

x > 4

shade right.

For:

x ≥ 4

shade right.

For:

x < 4

shade left.

For:

x ≤ 4

shade left.

A useful rule is:

greater → right

less → left

because values increase from left to right on the number line.

Graphing Negative Solutions

Suppose:

x ≤ −3

The graph has:

• closed circle at −3
• shading left

Remember:

−4 < −3

−5 < −3

−100 < −3

so these values all belong to the solution set.

From a Number Line to an Inequality

Suppose a graph has:

• open circle at 5
• shading right

The solution is:

x > 5

Suppose another graph has:

• closed circle at −2
• shading left

The solution is:

x ≤ −2

Always identify the boundary, inclusion, and direction.

Verifying Solutions by Substitution

After solving an inequality, you can test values from the proposed solution set.

Suppose:

2x < 10

Solution:

x < 5

Choose x = 3:

2(3) < 10

6 < 10

True.

Now choose a value outside the solution set, such as x = 7:

2(7) < 10

14 < 10

False.

This provides evidence that the solution direction is correct.

Checking the Boundary

If the solution is:

x ≤ 5

check x = 5.

If it makes the original inequality true, the closed circle is appropriate.

If the solution is:

x < 5

the boundary should not satisfy the original inequality.

Checking the boundary is particularly useful when deciding between:

< and ≤

or:

> and ≥

Checking a Negative-Division Problem

Solve:

−4x > 20

Divide by −4 and reverse:

x < −5

Check a solution such as x = −6:

−4(−6) > 20

24 > 20

True.

Check a non-solution such as x = 0:

−4(0) > 20

0 > 20

False.

The solution makes sense.

Understanding What a Solution Means

Consider:

x > 8

This does not mean:

x = 9

It means:

every number greater than 8

Examples include:

8.1, 9, 10, 100, 1000.

Similarly:

x ≤ −2

means every number less than or equal to −2.

The answer to an inequality describes a range of values.

Inequalities in Real Situations

One-step inequalities frequently represent real limits.

Examples include:

• budgets
• minimum ages
• maximum loads
• temperature limits
• minimum scores
• travel distances
• time limits

For example:

A student has $30 and has already spent $12.

If x represents additional spending:

x + 12 ≤ 30

Subtract 12:

x ≤ 18

The student can spend at most another $18.

Maximum Capacity Example

A container can hold at most:

50 kg

It already contains:

18 kg

If x represents additional mass:

x + 18 ≤ 50

Subtract 18:

x ≤ 32

The container can accept no more than another 32 kg.

In context, x would also normally satisfy:

x ≥ 0

Minimum Score Example

A student needs at least:

80 points

and already has:

63 points

If x represents the additional points required:

x + 63 ≥ 80

Subtract 63:

x ≥ 17

The student needs at least 17 additional points.

Temperature Example

A freezer begins at:

4°C

Its temperature must decrease by at least 10°C.

If x represents the temperature change:

x ≤ −10

The negative value indicates a decrease.

The final temperature would then be:

4 + x ≤ −6

This illustrates how negative quantities and inequalities can appear naturally in real situations.

Cost per Item Example

A student has:

$24

to spend on identical notebooks costing:

$4 each

If n represents the number of notebooks:

4n ≤ 24

Divide by 4:

n ≤ 6

Mathematically, any number ≤ 6 satisfies the inequality, but context tells us n must be a non-negative whole number.

Therefore:

n = 0, 1, 2, 3, 4, 5, or 6

Negative Rate Example

Suppose a quantity changes at:

−3 units per hour

and the total change must be less than:

−15 units.

If t is the number of hours:

−3t < −15

Divide by −3 and reverse:

t > 5

This example shows why reversing the inequality is essential.

Without reversing it, the interpretation would be incorrect.

Worked Example: Addition

Solve and graph:

x + 11 ≥ 18

Subtract 11:

x ≥ 7

Number line:

• closed circle at 7
• shade right

Check:

7 + 11 ≥ 18

18 ≥ 18

True.

Worked Example: Subtraction

Solve and graph:

x − 8 < −3

Add 8:

x < 5

Number line:

• open circle at 5
• shade left

Check x = 4:

4 − 8 < −3

−4 < −3

True.

Worked Example: Positive Multiplication

Solve:

7x ≥ 35

Divide by 7:

x ≥ 5

The symbol does not reverse because 7 is positive.

Check:

7(5) ≥ 35

35 ≥ 35

True.

Worked Example: Negative Multiplication

Solve:

−8x ≥ 32

Divide by −8.

Reverse ≥ to ≤:

x ≤ −4

Check:

−8(−4) ≥ 32

32 ≥ 32

True.

Worked Example: Positive Division

Solve:

x/6 < 4

Multiply by 6:

x < 24

Check x = 18:

18/6 < 4

3 < 4

True.

Worked Example: Negative Division

Solve:

x/−5 ≥ 6

Multiply by −5 and reverse:

x ≤ −30

Check:

−30/−5 ≥ 6

6 ≥ 6

True.

Comparing Two Similar Problems

Consider:

4x > 20

Divide by positive 4:

x > 5

Now:

−4x > 20

Divide by negative −4 and reverse:

x < −5

The negative coefficient changes both:

• the boundary value
• the direction of the inequality

A Reliable Solving Strategy

When solving a one-step inequality:

1. Identify the operation being performed on the variable.
2. Use the inverse operation on both sides.
3. Ask whether you multiplied or divided by a negative number.
4. If yes, reverse the inequality symbol.
5. Simplify the solution.
6. Identify the boundary value.
7. Decide whether the boundary is included.
8. Graph the solution on a number line.
9. Substitute a value to check the solution.
10. Interpret the solution in the context of the problem.

Common Mistakes

Treating an inequality exactly like an equation

The solving steps are similar, but inequalities usually represent many solutions.

Forgetting to reverse the inequality

If you multiply or divide by a negative number, reverse the symbol.

Reversing the symbol when adding or subtracting a negative number

Do not reverse it. The special rule applies only to multiplication and division by a negative.

Reversing when dividing by a positive number

The inequality direction stays unchanged.

Using an open circle for ≤ or ≥

These inequalities include the boundary, so use a closed circle.

Using a closed circle for < or >

These inequalities exclude the boundary, so use an open circle.

Shading in the wrong direction

Greater values are to the right. Smaller values are to the left.

Checking only the final inequality

Substitute into the original inequality when verifying your work.

Giving only one solution

For x > 5, writing x = 6 does not describe the entire solution set.

Ignoring context

A solution such as n ≤ 6 may mathematically include decimals and negative numbers, but n may represent a number of people or objects and therefore require non-negative whole numbers.

Key Terms

One-step inequality — inequality that can be solved using one inverse operation.

Inverse operation — operation that reverses another operation.

Solution — value that makes an inequality true.

Solution set — collection of all values that satisfy an inequality.

Boundary value — value separating solutions from non-solutions.

Substitution — replacing a variable with a value to determine whether a statement is true.

Open circle — number-line symbol showing that the boundary is excluded.

Closed circle — number-line symbol showing that the boundary is included.

Greater than (>) — indicates that one quantity is larger than another.

Less than (<) — indicates that one quantity is smaller than another.

Greater than or equal to (≥) — indicates that a value is greater than or equal to the boundary.

Less than or equal to (≤) — indicates that a value is less than or equal to the boundary.

Constraint — restriction placed on possible values.

Verification — process of checking whether a proposed solution satisfies the original inequality.

Key Takeaways

• One-step inequalities can be solved using inverse operations.
• Addition is reversed by subtraction.
• Subtraction is reversed by addition.
• Multiplication is reversed by division.
• Division is reversed by multiplication.
• Perform the same operation on both sides of the inequality.
• Adding or subtracting a number does not reverse the inequality.
• Multiplying or dividing by a positive number does not reverse the inequality.
• Multiplying or dividing by a negative number reverses the inequality symbol.
• The reversal occurs because multiplication by a negative reverses the order of numbers on a number line.
• Inequality solutions usually represent ranges rather than individual numbers.
• Open circles represent excluded boundary values.
• Closed circles represent included boundary values.
• Greater-than solutions extend right on a number line.
• Less-than solutions extend left.
• Solutions can be verified by substituting values into the original inequality.
• Checking both a solution and a non-solution can help confirm the direction of an inequality.
• Real-world context can restrict a mathematical solution set.
• A complete solution should include the inequality, graphical representation when appropriate, and interpretation of what the solution means.

Check Your Understanding

1. Solve: x + 7 > 15. Describe its number-line graph.
2. Solve: x − 9 ≤ 4. Describe its number-line graph.
3. Solve: x + 12 ≥ 5.
4. Solve: x − 6 < −10.
5. Solve: 5x > 30 and verify your answer using one value from the solution set.
6. Solve: 8x ≤ 48 and explain whether the boundary value is included.
7. Solve: x/4 ≥ 7 and describe its number-line graph.
8. Solve: x/6 < −3.
9. Solve: −3x > 18. Explain why the inequality symbol must reverse.
10. Solve: −7x ≤ 35 and verify the boundary value by substitution.
11. Solve: x/−4 > 5. Describe its number-line graph.
12. A student claims that −5x < 20 gives x < −4. Identify the mistake and give the correct solution.
13. A container has a maximum capacity of 80 kg and already contains 27 kg. Write and solve an inequality for the additional mass x that can be added. Explain what the solution means.
14. A student needs at least 75 points to pass and already has 58 points. Write and solve an inequality representing the additional points x needed.
15. Solve each inequality and compare the results: 4x ≥ 20 and −4x ≥ 20. Explain why the solution directions are different.

Solving Multi-Step Inequalities

A multi-step inequality requires more than one operation to isolate the variable. Solving these inequalities combines several algebra skills:

• simplifying expressions
• combining like terms
• using the distributive property
• performing inverse operations
• working with variables on both sides
• reversing the inequality when multiplying or dividing by a negative number
• representing the final solution using a number line and interval notation

For example:

3x + 5 < 20

Subtract 5:

3x < 15

Divide by 3:

x < 5

The solution represents all real numbers less than 5, not just one value.

Review of Inequality Symbols

The four main inequality symbols are:

The symbols ≤ and ≥ include the boundary value.

For example:

x < 4

does not include 4.

But:

x ≤ 4

does include 4.

Solving Inequalities Like Equations

Most algebraic operations used to solve equations also work with inequalities.

You can:

• add the same number to both sides
• subtract the same number from both sides
• multiply both sides by the same positive number
• divide both sides by the same positive number

For example:

2x + 7 > 15

Subtract 7:

2x > 8

Divide by 2:

x > 4

The inequality direction remains unchanged because we divided by a positive number.

The Negative-Number Rule

There is one critical difference between equations and inequalities:

When multiplying or dividing both sides by a negative number, reverse the inequality symbol.

For example:

−3x > 12

Divide by −3.

Because −3 is negative:

> becomes <

Therefore:

x < −4

This rule is essential when solving multi-step inequalities.

 

 

 

7

Why the Inequality Reverses

Start with a true statement:

2 < 5

Multiply both sides by −1:

−2 and −5

On the number line:

−2 > −5

Therefore:

2 < 5

becomes:

−2 > −5

Multiplication by a negative reflects numbers across zero and reverses their order.

The same effect occurs when dividing by a negative number.

Combining Like Terms

Before solving an inequality, simplify each side whenever possible.

Like terms have the same variable raised to the same power.

For example:

3x + 5x = 8x

and:

7 − 2 = 5

Consider:

3x + 2x + 4 > 19

Combine like terms:

5x + 4 > 19

Subtract 4:

5x > 15

Divide by 5:

x > 3

Combining Constants

Consider:

6x + 7 − 3 > 28

Combine the constants:

6x + 4 > 28

Subtract 4:

6x > 24

Divide by 6:

x > 4

Simplifying first makes the remaining steps easier to see.

Combining Several Like Terms

Solve:

4x + 3 + 2x − 7 ≤ 20

Combine variable terms:

4x + 2x = 6x

Combine constants:

3 − 7 = −4

So:

6x − 4 ≤ 20

Add 4:

6x ≤ 24

Divide by 6:

x ≤ 4

Order of Operations

When simplifying an inequality, follow the usual order of operations.

A useful sequence is:

1. simplify parentheses
2. combine like terms on each side
3. move variable terms if necessary
4. move constants
5. isolate the variable
6. check whether the final multiplication or division uses a negative number
7. reverse the symbol if required

This systematic approach reduces errors.

Two-Step Inequalities

Consider:

4x − 7 ≥ 13

Add 7:

4x ≥ 20

Divide by 4:

x ≥ 5

Number line:

• closed circle at 5
• shade right

Interval notation:

[5, ∞)

Another Two-Step Example

Solve:

5x + 8 < 28

Subtract 8:

5x < 20

Divide by 5:

x < 4

Number line:

• open circle at 4
• shade left

Interval notation:

(−∞, 4)

Multi-Step Inequality with a Negative Coefficient

Solve:

−4x + 7 > 19

Subtract 7:

−4x > 12

Divide by −4.

Reverse > to <:

x < −3

Number line:

• open circle at −3
• shade left

Interval notation:

(−∞, −3)

Another Negative-Coefficient Example

Solve:

−5x − 8 ≤ 17

Add 8:

−5x ≤ 25

Divide by −5.

Reverse ≤ to ≥:

x ≥ −5

Interval notation:

[−5, ∞)

The Distributive Property

Some multi-step inequalities contain parentheses.

Use the distributive property:

a(b + c) = ab + ac

For example:

3(x + 4) > 21

Distribute:

3x + 12 > 21

Subtract 12:

3x > 9

Divide by 3:

x > 3

Distributing a Negative Number

Extra care is needed when the number outside the parentheses is negative.

Solve:

−2(x + 3) ≥ 10

Distribute:

−2x − 6 ≥ 10

Add 6:

−2x ≥ 16

Divide by −2 and reverse:

x ≤ −8

The inequality reverses during the division by −2, not simply because a negative number appeared earlier.

Distributing Across Subtraction

Solve:

4(2x − 3) < 20

Distribute:

8x − 12 < 20

Add 12:

8x < 32

Divide by 8:

x < 4

Combining Like Terms After Distributing

Solve:

2(x + 5) + 3x > 25

Distribute:

2x + 10 + 3x > 25

Combine like terms:

5x + 10 > 25

Subtract 10:

5x > 15

Divide by 5:

x > 3

Variables on Both Sides

Multi-step inequalities may contain variables on both sides.

Consider:

5x + 3 > 2x + 12

Subtract 2x from both sides:

3x + 3 > 12

Subtract 3:

3x > 9

Divide by 3:

x > 3

The goal is to collect variable terms on one side.

Choosing Which Variable Term to Move

Consider:

3x + 10 ≤ 7x − 6

We could subtract 3x:

10 ≤ 4x − 6

Add 6:

16 ≤ 4x

Divide by 4:

4 ≤ x

Usually it is clearer to write the variable first:

x ≥ 4

Notice that:

4 ≤ x

and:

x ≥ 4

mean exactly the same thing.

Avoiding Negative Coefficients

Sometimes choosing which variable term to move can make the problem easier.

Consider:

2x + 8 > 5x − 4

Subtract 2x:

8 > 3x − 4

Add 4:

12 > 3x

Divide by 3:

4 > x

Rewrite:

x < 4

This approach avoids creating a negative coefficient.

Example Producing a Negative Coefficient

The same problem can be solved another way:

2x + 8 > 5x − 4

Subtract 5x:

−3x + 8 > −4

Subtract 8:

−3x > −12

Divide by −3 and reverse:

x < 4

Both methods give the same answer.

This demonstrates that different correct algebraic approaches should produce the same solution set.

Variables on Both Sides with a Negative Result

Solve:

7x + 4 ≤ 3x − 12

Subtract 3x:

4x + 4 ≤ −12

Subtract 4:

4x ≤ −16

Divide by 4:

x ≤ −4

Interval notation:

(−∞, −4]

Simplifying Both Sides

Consider:

3x + 4 + 2x > x + 16

Combine like terms on the left:

5x + 4 > x + 16

Subtract x:

4x + 4 > 16

Subtract 4:

4x > 12

Divide by 4:

x > 3

Parentheses on Both Sides

Solve:

2(x + 3) ≤ 4(x − 1)

Distribute:

2x + 6 ≤ 4x − 4

Subtract 2x:

6 ≤ 2x − 4

Add 4:

10 ≤ 2x

Divide by 2:

5 ≤ x

Therefore:

x ≥ 5

Interval notation:

[5, ∞)

A More Challenging Example

Solve:

−3(2x − 4) > 2(x + 5)

Distribute:

−6x + 12 > 2x + 10

Subtract 2x:

−8x + 12 > 10

Subtract 12:

−8x > −2

Divide by −8 and reverse:

x < 1/4

Interval notation:

(−∞, 1/4)

Inequalities with Fractions

Fractions do not change the basic method.

Solve:

x/3 + 4 > 9

Subtract 4:

x/3 > 5

Multiply by 3:

x > 15

Because 3 is positive, the inequality does not reverse.

Clearing Fractions

Consider:

x/4 + x/2 ≤ 9

The least common denominator is 4.

Multiply every term by 4:

x + 2x ≤ 36

3x ≤ 36

Divide by 3:

x ≤ 12

Interval notation:

(−∞, 12]

Clearing Fractions with a Negative Multiplier

Be careful if you choose to multiply an inequality by a negative number.

For example:

−x/3 > 4

Multiply by −3.

Because −3 is negative, reverse the symbol:

x < −12

A safer alternative is sometimes to first multiply by positive 3:

−x > 12

Then divide by −1 and reverse:

x < −12

Decimal Inequalities

Decimals can be solved in the same way.

Solve:

0.5x + 2 > 7

Subtract 2:

0.5x > 5

Divide by 0.5:

x > 10

You can also multiply the entire inequality by 2:

x + 4 > 14

x > 10

Interval Notation

Interval notation is a compact way to represent a set of numbers.

For example:

x > 3

can be written:

(3, ∞)

This means all real numbers greater than 3.

 

 

 

8

Parentheses in Interval Notation

Parentheses indicate that an endpoint is not included.

For:

x > 5

interval notation is:

(5, ∞)

For:

x < 5

interval notation is:

(−∞, 5)

These correspond to open circles on a number line.

Square Brackets in Interval Notation

Square brackets indicate that an endpoint is included.

For:

x ≥ 5

interval notation is:

[5, ∞)

For:

x ≤ 5

interval notation is:

(−∞, 5]

These correspond to closed circles at the finite endpoint.

Infinity Always Uses Parentheses

Infinity is not a number that can be reached or included.

Therefore:

∞ and −∞ always use parentheses.

Correct:

[4, ∞)

Correct:

(−∞, 7]

Incorrect:

[4, ∞]

Incorrect:

[−∞, 7]

Inequality, Number Line, and Interval Notation

These are three different ways of representing the same solution set.

Representing a Solution on a Number Line

Suppose the solution is:

x ≤ −2

The number-line graph has:

• a closed circle at −2
• shading extending left

Interval notation:

(−∞, −2]

Now consider:

x > −2

The graph has:

• an open circle at −2
• shading extending right

Interval notation:

(−2, ∞)

Verifying by Substitution

Substitution can be used to check whether particular values satisfy the original inequality.

Suppose:

3x + 4 < 19

We solve:

3x < 15

x < 5

Choose a value from the solution set:

x = 4

Substitute into the original inequality:

3(4) + 4 < 19

16 < 19

True.

Therefore, x = 4 is a valid solution.

Testing a Value Outside the Solution Set

Using the same inequality:

3x + 4 < 19

with solution:

x < 5

try:

x = 6

3(6) + 4 < 19

22 < 19

False.

This is exactly what we expect because 6 is outside the solution set.

Testing the Boundary

For:

3x + 4 < 19

the boundary is:

x = 5

Substitute:

3(5) + 4 < 19

19 < 19

False.

Therefore, 5 is correctly excluded.

If the original inequality had been:

3x + 4 ≤ 19

then:

19 ≤ 19

would be true, so 5 would be included.

Why Testing One Value Is Not a Complete Proof

Testing one number can show that a particular number works, but it does not prove that every number in the proposed interval works.

The algebraic solution establishes the complete solution set.

Substitution is useful for:

• checking the boundary
• checking a typical solution
• detecting a reversed inequality
• finding arithmetic mistakes

It should support the algebra rather than replace it.

Special Case: All Real Numbers

Sometimes the variable disappears during simplification.

Consider:

2(x + 3) < 2x + 10

Distribute:

2x + 6 < 2x + 10

Subtract 2x:

6 < 10

This statement is always true.

Therefore, every real number is a solution.

We can write:

all real numbers

Interval notation:

(−∞, ∞)

Why Every Number Works

Test any value.

For x = 100:

2(100 + 3) < 2(100) + 10

206 < 210

True.

For x = −50:

2(−50 + 3) < 2(−50) + 10

−94 < −90

True.

The inequality is true regardless of x.

Special Case: No Solution

Consider:

3(x + 2) > 3x + 10

Distribute:

3x + 6 > 3x + 10

Subtract 3x:

6 > 10

This statement is always false.

Therefore:

no solution

No real value of x can make the original inequality true.

Recognizing Special Cases

If the variable disappears, examine the remaining statement.

If the statement is true:

all real numbers

Example:

4 < 9

If the statement is false:

no solution

Example:

12 < 5

Do not try to continue solving for x after the variable has disappeared.

Number-Line Representation of Special Cases

For all real numbers, the entire number line is shaded.

Interval notation:

(−∞, ∞)

For no solution, no part of the number line is shaded.

The solution set may also be represented by:

∅

which means the empty set.

Real-World Multi-Step Inequalities

Multi-step inequalities are useful for situations involving:

• budgets
• fees
• wages
• transportation
• production
• capacity
• temperature ranges
• minimum requirements

For example, suppose a taxi charges:

• $5 starting fee
• $2 per kilometre

A passenger has no more than $25.

If d is the distance:

5 + 2d ≤ 25

Subtract 5:

2d ≤ 20

Divide by 2:

d ≤ 10

The passenger can travel at most 10 km under this simplified model.

Budget Example

A student has:

$80

to spend.

There is a fixed delivery charge of:

$8

and each item costs:

$12.

If n is the number of items:

12n + 8 ≤ 80

Subtract 8:

12n ≤ 72

Divide by 12:

n ≤ 6

Because n represents a number of items, realistic solutions are non-negative whole numbers up to 6.

Minimum Earnings Example

A worker receives:

$50

as a fixed payment plus:

$18 per hour.

The worker wants to earn at least:

$230.

Let h be the number of hours:

50 + 18h ≥ 230

Subtract 50:

18h ≥ 180

Divide by 18:

h ≥ 10

The worker must work at least 10 hours.

Comparing Two Plans

Suppose Plan A costs:

20 + 5x

and Plan B costs:

50 + 2x

We want to know when Plan A is cheaper:

20 + 5x < 50 + 2x

Subtract 2x:

20 + 3x < 50

Subtract 20:

3x < 30

Divide by 3:

x < 10

Plan A is cheaper when x is less than 10, assuming x represents values allowed by the context.

A Complete Multi-Step Example

Solve:

4(2x − 3) + 5 ≤ 3x + 18

Distribute:

8x − 12 + 5 ≤ 3x + 18

Combine like terms:

8x − 7 ≤ 3x + 18

Subtract 3x:

5x − 7 ≤ 18

Add 7:

5x ≤ 25

Divide by 5:

x ≤ 5

Interval notation:

(−∞, 5]

Number line:

• closed circle at 5
• shade left

Verification using x = 5:

4[2(5) − 3] + 5 ≤ 3(5) + 18

4(7) + 5 ≤ 33

33 ≤ 33

True.

Complete Example with a Negative Coefficient

Solve:

3(x − 4) − 5x > 8

Distribute:

3x − 12 − 5x > 8

Combine:

−2x − 12 > 8

Add 12:

−2x > 20

Divide by −2 and reverse:

x < −10

Interval notation:

(−∞, −10)

Check x = −11:

3(−11 − 4) − 5(−11) > 8

3(−15) + 55 > 8

−45 + 55 > 8

10 > 8

True.

Complete Example with Variables on Both Sides

Solve:

5(x + 2) ≤ 3(x + 6)

Distribute:

5x + 10 ≤ 3x + 18

Subtract 3x:

2x + 10 ≤ 18

Subtract 10:

2x ≤ 8

Divide by 2:

x ≤ 4

Interval notation:

(−∞, 4]

Check x = 4:

5(4 + 2) ≤ 3(4 + 6)

30 ≤ 30

True.

Complete Example with Fractions

Solve:

x/2 + x/4 − 3 > 6

Multiply every term by 4:

2x + x − 12 > 24

Combine:

3x − 12 > 24

Add 12:

3x > 36

Divide by 3:

x > 12

Interval notation:

(12, ∞)

Complete Example Producing All Real Numbers

Solve:

4(x − 2) + 3 < 4x

Distribute:

4x − 8 + 3 < 4x

Combine:

4x − 5 < 4x

Subtract 4x:

−5 < 0

This is always true.

Therefore:

all real numbers

Interval notation:

(−∞, ∞)

Complete Example Producing No Solution

Solve:

2(x + 5) > 2x + 14

Distribute:

2x + 10 > 2x + 14

Subtract 2x:

10 > 14

This is false.

Therefore:

no solution

or:

∅

A Reliable Multi-Step Strategy

When solving a multi-step inequality:

1. Simplify both sides.
   Distribute and combine like terms.
2. Move variable terms to one side.
   Add or subtract variable terms.
3. Move constants to the other side.
   Use inverse operations.
4. Isolate the variable.
   Multiply or divide as necessary.
5. Check for a negative multiplication or division.
   Reverse the inequality if required.
6. Write the solution clearly.
7. Convert to interval notation.
8. Represent it on a number line.
9. Verify using substitution.
10. Interpret the answer in context if the problem describes a real situation.

Common Mistakes

Not simplifying first

Combining like terms early usually makes the inequality easier to solve.

Distributing incorrectly

For:

−3(x − 4)

the correct result is:

−3x + 12

The negative must multiply every term.

Combining unlike terms

For example:

3x + 5

cannot be simplified to 8x.

Forgetting to perform operations on both sides

An inequality must remain balanced just like an equation.

Reversing the inequality when adding or subtracting

The sign reverses only when multiplying or dividing by a negative number.

Forgetting to reverse after dividing by a negative

For:

−2x > 8

the correct solution is:

x < −4

Reversing because a negative number appears somewhere

A negative number appearing in the inequality does not automatically cause reversal.

Using brackets with infinity

Infinity always uses parentheses.

Confusing parentheses and brackets

Parentheses exclude a finite endpoint. Square brackets include it.

Forgetting special cases

If the variable disappears, the answer may be all real numbers or no solution.

Testing only the simplified inequality

When verifying, substitute into the original inequality.

Key Terms

Multi-step inequality — inequality requiring several algebraic operations to isolate the variable.

Like terms — terms with identical variable parts that can be combined.

Distributive property — rule used to multiply a factor across terms inside parentheses.

Inverse operation — operation used to undo another operation.

Solution set — collection of all values satisfying an inequality.

Boundary value — value separating solutions from non-solutions.

Interval notation — compact notation used to describe continuous sets of real numbers.

Parenthesis — interval symbol showing that a finite endpoint is excluded.

Square bracket — interval symbol showing that a finite endpoint is included.

Infinity — concept describing values extending without bound; always written with a parenthesis in interval notation.

Substitution — replacing a variable with a particular value.

Verification — checking whether a proposed solution satisfies the original inequality.

All real numbers — solution set containing every real number, written (−∞, ∞).

No solution — situation in which no value satisfies an inequality.

Empty set (∅) — notation representing a solution set containing no elements.

Key Takeaways

• Multi-step inequalities combine several algebraic skills.
• Simplify expressions before isolating the variable.
• Use the distributive property correctly when parentheses are present.
• Combine only like terms.
• Variables may need to be moved from one side of the inequality to the other.
• Addition and subtraction do not reverse the inequality.
• Multiplication or division by a positive number does not reverse the inequality.
• Multiplication or division by a negative number reverses the inequality.
• Different correct algebraic methods should produce the same solution set.
• Solutions can be represented using inequalities, number lines, and interval notation.
• Open endpoints correspond to < or >.
• Closed endpoints correspond to ≤ or ≥.
• Infinity and negative infinity always use parentheses.
• Substitution can help verify a solution.
• Checking the boundary is especially useful for determining whether it should be included.
• If the variable disappears and a true statement remains, the solution is all real numbers.
• If the variable disappears and a false statement remains, there is no solution.
• Real-world contexts may further restrict a mathematical solution set.
• A complete solution should clearly communicate the algebra, final inequality, interval notation, and graphical interpretation.

Check Your Understanding

1. Simplify and solve: 3x + 5x − 4 > 20. Write the answer in interval notation.
2. Solve: 5x + 7 ≤ 32. Describe its number-line representation.
3. Solve: −4x + 6 > 22. Explain exactly when and why the inequality reverses.
4. Solve: 3(x + 5) ≥ 27. Give the inequality and interval notation.
5. Solve: −2(x − 4) < 16. Verify your answer using substitution.
6. Solve: 5x + 8 > 2x + 20. Represent the solution on a number line and in interval notation.
7. Solve: 3x − 7 ≤ 6x + 8.
8. Solve: 4(x − 2) + 3x > 20.
9. Solve: −3(2x + 5) ≥ 4x − 5. Write the answer in interval notation.
10. Solve: x/3 + x/6 ≤ 8.
11. Solve: 2(x + 4) < 2x + 12. Explain why the answer is all real numbers.
12. Solve: 5(x − 1) > 5x + 2. Explain why the inequality has no solution.
13. A taxi charges a $6 starting fee plus $3 per kilometre. You have no more than $42. Write and solve an inequality to determine the maximum distance you can travel.
14. Plan A costs $25 + $4x, while Plan B costs $55 + $2x. Determine when Plan A is cheaper than Plan B. Express the answer as an inequality and interval.
15. Solve and fully communicate the solution to 4(2x − 3) − 5x ≤ −2(x + 6) + 9. Show the simplification, explain any inequality reversal if needed, give the final inequality, write the solution in interval notation, describe its number-line graph, and verify one value by substitution.

Compound Inequalities

A compound inequality combines two inequalities to describe a solution set with more than one condition.

For example:

x > 2 and x < 7

means that x must satisfy both conditions.

The solution contains numbers greater than 2 but less than 7:

2 < x < 7

Another example is:

x < −3 or x > 4

Here, x only needs to satisfy one of the two conditions.

Understanding the difference between and and or is the key to working with compound inequalities.

What Makes an Inequality Compound?

A simple inequality contains one comparison:

x > 5

A compound inequality combines two comparisons:

x > 5 and x < 10

or:

x < −2 or x ≥ 6

Compound inequalities are useful whenever a quantity has:

• both a lower and upper limit
• two separate acceptable ranges
• multiple conditions that must be satisfied

The Meaning of "And"

An and inequality requires both conditions to be true at the same time.

Consider:

x > 2 and x < 8

For a number to be a solution, it must be:

• greater than 2
• less than 8

Therefore:

2 < x < 8

Examples of solutions include:

3, 4, 5, 6, 7, 2.5, and 7.9.

But 2 and 8 are not included.

 

 

 

9

"And" Means Intersection

The solution to an and inequality is the overlap of the two individual solution sets.

For:

x > 2 and x < 8

the first condition gives numbers to the right of 2.

The second gives numbers to the left of 8.

The overlap is:

2 < x < 8

This overlap is called the intersection.

A useful way to remember this is:

AND = OVERLAP

Writing an "And" Inequality as One Statement

The statement:

x ≥ 3 and x < 9

can be written:

3 ≤ x < 9

Read this as:

"x is greater than or equal to 3 and less than 9."

The boundary 3 is included.

The boundary 9 is not included.

Interval notation:

[3, 9)

Double Inequalities

Expressions such as:

2 < x < 10

are called double inequalities.

The expression really contains two statements:

2 < x

and:

x < 10

Together:

x > 2 and x < 10

The variable lies between two boundaries.

Reading Double Inequalities

Consider:

−4 ≤ x < 6

This means:

x ≥ −4 and x < 6

Therefore:

• −4 is included
• 6 is not included
• every number between them is included

Interval notation:

[−4, 6)

The Meaning of "Or"

An or inequality requires at least one of the conditions to be true.

Consider:

x < −2 or x > 5

The solution includes:

• all numbers less than −2
• all numbers greater than 5

Examples include:

−10, −3, 6, 20, and 100.

Numbers between −2 and 5 do not satisfy either condition.

 

 

 

7

"Or" Means Union

For:

x < −2 or x > 5

the solution combines two separate regions.

This is called the union of the solution sets.

A useful way to remember this is:

OR = COMBINE

"And" usually looks for overlap.

"Or" usually combines the acceptable regions.

Comparing "And" and "Or"

Consider these two statements:

x > 2 and x < 7

Solution:

2 < x < 7

Now:

x < 2 or x > 7

Solution:

all numbers outside the interval from 2 to 7.

Graphically, these have very different appearances.

 

 

 

8

Testing an "And" Statement

Consider:

x ≥ 4 and x < 10

Test:

x = 6

First condition:

6 ≥ 4

True.

Second condition:

6 < 10

True.

Both are true, so:

x = 6 is a solution.

Now test x = 12.

12 ≥ 4 is true.

12 < 10 is false.

Because both conditions must be true:

x = 12 is not a solution.

Testing an "Or" Statement

Consider:

x < −3 or x ≥ 5

Test:

x = 8

First condition:

8 < −3

False.

Second condition:

8 ≥ 5

True.

Because at least one condition is true:

x = 8 is a solution.

For an "or" statement, both conditions do not need to be true.

Number Lines for "And" Inequalities

Consider:

−2 < x ≤ 5

The number-line representation has:

• open circle at −2
• closed circle at 5
• shading between the two boundaries

The open circle means −2 is excluded.

The closed circle means 5 is included.

Interval notation:

(−2, 5]

Number Lines for "Or" Inequalities

Consider:

x ≤ −2 or x > 5

The graph has:

• closed circle at −2 and shading left
• open circle at 5 and shading right

There are two separate regions.

Interval notation:

(−∞, −2] ∪ (5, ∞)

The symbol ∪ means union.

Interval Notation Review

Interval notation uses parentheses and square brackets.

means the endpoint is not included.

[ ] means the endpoint is included.

For example:

2 < x ≤ 8

becomes:

(2, 8]

The 2 uses a parenthesis because it is excluded.

The 8 uses a bracket because it is included.

Infinity in Interval Notation

Infinity always uses parentheses.

For:

x < 4

interval notation is:

(−∞, 4)

For:

x ≥ 4

interval notation is:

[4, ∞)

We never write:

[∞]

because infinity is not an actual endpoint that can be included.

Union Notation

When an "or" inequality produces two separate intervals, use the union symbol:

∪

For example:

x < −1 or x ≥ 6

becomes:

(−∞, −1) ∪ [6, ∞)

This means:

the first interval or the second interval.

Solving an "And" Compound Inequality

Consider:

x + 3 > 5 and x + 3 < 11

Solve each condition.

First:

x + 3 > 5

x > 2

Second:

x + 3 < 11

x < 8

Combine:

2 < x < 8

Interval notation:

(2, 8)

Solving a Double Inequality Together

A double inequality can often be solved as one expression.

Consider:

3 < x + 2 ≤ 9

Subtract 2 from all three parts:

3 − 2 < x + 2 − 2 ≤ 9 − 2

Therefore:

1 < x ≤ 7

Interval notation:

(1, 7]

This is often more efficient than separating the inequality into two statements.

Performing Operations on All Three Parts

Consider:

−4 ≤ 2x < 10

Divide all three parts by 2:

−2 ≤ x < 5

Therefore:

−2 ≤ x < 5

Interval notation:

[−2, 5)

When solving a double inequality, whatever operation is performed must be applied throughout the entire inequality.

Multiplying a Double Inequality by a Negative

The negative-number rule still applies.

Consider:

−8 < −2x ≤ 6

Divide every part by −2.

Because we are dividing by a negative number, both inequality symbols reverse:

4 > x ≥ −3

It is clearer to rewrite this from smallest to largest:

−3 ≤ x < 4

Interval notation:

[−3, 4)

Why Both Symbols Reverse

Start with:

−8 < −2x ≤ 6

After division by −2:

−8/−2 > x ≥ 6/−2

giving:

4 > x ≥ −3

Reordering from least to greatest gives:

−3 ≤ x < 4

Every comparison reverses because every part of the inequality was divided by a negative number.

Solving a Multi-Step Double Inequality

Solve:

5 ≤ 2x + 1 < 13

Subtract 1 from all three parts:

4 ≤ 2x < 12

Divide all parts by 2:

2 ≤ x < 6

Interval notation:

[2, 6)

Number line:

• closed at 2
• open at 6
• shade between them

Another Multi-Step Example

Solve:

−7 < 3x + 2 ≤ 14

Subtract 2:

−9 < 3x ≤ 12

Divide by 3:

−3 < x ≤ 4

Interval notation:

(−3, 4]

Solving "Or" Inequalities Separately

"Or" inequalities are normally solved as two separate inequalities.

Consider:

2x < −6 or 3x ≥ 12

Solve the first:

2x < −6

x < −3

Solve the second:

3x ≥ 12

x ≥ 4

Therefore:

x < −3 or x ≥ 4

Interval notation:

(−∞, −3) ∪ [4, ∞)

"Or" Example with Negative Coefficients

Solve:

−3x > 9 or 2x ≥ 10

First inequality:

−3x > 9

Divide by −3 and reverse:

x < −3

Second inequality:

2x ≥ 10

x ≥ 5

Therefore:

x < −3 or x ≥ 5

Interval notation:

(−∞, −3) ∪ [5, ∞)

Compound Inequalities with Variables on Both Sides

Consider:

3x + 2 > x + 6 and 2x − 1 ≤ 9

Solve the first:

3x + 2 > x + 6

2x + 2 > 6

2x > 4

x > 2

Solve the second:

2x − 1 ≤ 9

2x ≤ 10

x ≤ 5

Both conditions must be true:

2 < x ≤ 5

Interval notation:

(2, 5]

Finding the Overlap

Suppose we solve an "and" inequality and obtain:

x ≥ −1

and:

x < 4

The overlap is:

−1 ≤ x < 4

But suppose we obtain:

x > 6

and:

x < 2

No number can satisfy both conditions.

Therefore:

no solution

or:

∅

"And" with No Solution

Consider:

x + 2 > 10 and x − 3 < 2

First:

x > 8

Second:

x < 5

The compound condition requires:

x > 8 and x < 5

No number satisfies both.

Therefore:

∅

The number line has no shaded region.

Overlapping "Or" Solutions

Consider:

x > 2 or x > 6

Every number greater than 6 is already included in:

x > 2

Therefore, the combined solution is simply:

x > 2

Interval notation:

(2, ∞)

We do not need to write two intervals when one completely contains the other.

An "Or" Inequality Giving All Real Numbers

Consider:

x < 5 or x > 2

Think about the two solution sets.

The first contains everything below 5.

The second contains everything above 2.

Together, they cover every real number.

Therefore:

all real numbers

Interval notation:

(−∞, ∞)

Visualizing Overlap

Number lines are especially useful for compound inequalities.

For an "and" problem:

1. graph each condition
2. identify where the graphs overlap
3. keep only the overlap

For an "or" problem:

1. graph each condition
2. combine every region covered by either graph

This makes the logical difference between "and" and "or" visible.

Practical Situation: Temperature Range

A laboratory sample must remain between:

2°C and 8°C, inclusive.

Let T represent temperature.

Because both boundaries are included:

2 ≤ T ≤ 8

Interval notation:

[2, 8]

This is an and situation because the temperature must satisfy both the minimum and maximum conditions.

Practical Situation: Safe Operating Temperature

Suppose a machine operates safely only when its temperature is:

• at least 20°C
• below 70°C

Then:

20 ≤ T < 70

Interval notation:

[20, 70)

The machine must satisfy both conditions simultaneously.

Practical Situation: Age Range

A program accepts students who are:

• at least 12 years old
• no older than 16

Let a represent age.

Then:

12 ≤ a ≤ 16

Interval notation:

[12, 16]

Practical Situation: Values Outside a Range

Suppose a warning system activates when temperature is:

• below 0°C
• or above 100°C

Then:

T < 0 or T > 100

Interval notation:

(−∞, 0) ∪ (100, ∞)

This is an "or" condition because either extreme activates the warning.

Practical Situation: Manufacturing Tolerance

A component should have a length of at least:

49.8 mm

and at most:

50.2 mm

Let L represent length.

Then:

49.8 ≤ L ≤ 50.2

Interval notation:

[49.8, 50.2]

This type of compound inequality is extremely important in manufacturing because real components must remain within acceptable tolerances.

 

 

 

9

Practical Situation: Speed Range

Suppose a machine operates correctly only between:

1200 rpm and 1800 rpm

including both endpoints.

Let n represent rotational speed.

Then:

1200 ≤ n ≤ 1800

The acceptable interval is:

[1200, 1800]

A speed of 1500 rpm is acceptable.

A speed of 2000 rpm is not.

Building a Compound Inequality from Words

Consider:

"A number is greater than −4 but no more than 10."

"Greater than −4" means:

x > −4

"No more than 10" means:

x ≤ 10

Because both conditions must hold:

−4 < x ≤ 10

Interval notation:

(−4, 10]

Understanding "Between"

The word between can sometimes be ambiguous in everyday language.

For example:

"x is between 3 and 8"

might mean:

3 < x < 8

But if the endpoints are included, the wording should make this clear:

"x is between 3 and 8, inclusive"

means:

3 ≤ x ≤ 8

Always check whether the boundary values are allowed.

Understanding "Outside"

Suppose:

"x lies outside the interval from 2 to 7."

If the boundaries are also outside the acceptable region, this could be written:

x < 2 or x > 7

If the boundaries are included in the outside region:

x ≤ 2 or x ≥ 7

The exact inequality depends on whether the boundary values belong to the solution set.

Solving a Practical Multi-Step Problem

A freezer should operate between:

−20°C and −12°C

inclusive.

A sensor reports a transformed value:

T = 2x − 4

We require:

−20 ≤ 2x − 4 ≤ −12

Add 4:

−16 ≤ 2x ≤ −8

Divide by 2:

−8 ≤ x ≤ −4

Interval notation:

[−8, −4]

Solving a Practical "Or" Problem

A machine automatically shuts down when a sensor value is:

less than −10 or greater than 15

Suppose the sensor value is:

2x + 3

Then:

2x + 3 < −10 or 2x + 3 > 15

Solve the first:

2x < −13

x < −6.5

Solve the second:

2x > 12

x > 6

Therefore:

x < −6.5 or x > 6

Interval notation:

(−∞, −6.5) ∪ (6, ∞)

Verifying an "And" Solution

Suppose:

−2 ≤ x < 5

Choose:

x = 3

Check:

−2 ≤ 3

True.

3 < 5

True.

Both conditions are true.

Therefore, 3 is a solution.

Now test x = 6:

−2 ≤ 6

True.

6 < 5

False.

Therefore, 6 is not a solution.

Verifying an "Or" Solution

Suppose:

x < −4 or x ≥ 7

Test x = 10:

10 < −4

False.

10 ≥ 7

True.

Because one condition is true, 10 is a solution.

Test x = 2:

2 < −4

False.

2 ≥ 7

False.

Neither condition is true, so 2 is not a solution.

Checking Boundary Values

Boundary values are particularly important with compound inequalities.

Consider:

−3 < x ≤ 8

Test x = −3:

−3 < −3

False.

So −3 is excluded.

Test x = 8:

8 ≤ 8

True.

So 8 is included.

This agrees with:

(−3, 8]

A Reliable Strategy for "And" Inequalities

When solving an "and" inequality:

1. Identify both conditions.
2. Solve each inequality.
3. Reverse inequality signs if multiplying or dividing by negatives.
4. Find the values satisfying both conditions.
5. Identify the overlap.
6. Write the final inequality.
7. Convert to interval notation.
8. Graph the solution.
9. Check important values, especially boundaries.

Remember:

AND = OVERLAP

A Reliable Strategy for "Or" Inequalities

When solving an "or" inequality:

1. Separate the two inequalities.
2. Solve each one independently.
3. Reverse signs when required.
4. Keep values satisfying either condition.
5. Combine the solution sets.
6. Use ∪ when separate intervals remain.
7. Graph both regions.
8. Check boundary values.

Remember:

OR = COMBINE

Common Mistakes

Confusing "and" with "or"

"And" requires both conditions. "Or" requires at least one.

Graphing an "and" solution as two outward rays

"And" frequently produces the region between two boundaries.

Graphing an "or" solution only between the boundaries

"Or" often produces two separate regions outside a central interval.

Forgetting to reverse inequality signs

If you multiply or divide by a negative number, reverse the inequality.

Reversing only one symbol in a double inequality

If every part is divided or multiplied by a negative number, both inequality symbols reverse.

Forgetting to operate on all three parts

For:

2 < x + 5 ≤ 10

subtract 5 from all three parts.

Using the wrong interval symbol

Use parentheses for excluded finite boundaries and brackets for included finite boundaries.

Using brackets with infinity

Infinity always uses parentheses.

Forgetting the union symbol

Separate "or" intervals are joined using:

∪

Failing to simplify overlapping intervals

For:

x > 2 or x > 5

the final solution is simply:

x > 2

Assuming every "and" inequality has a solution

Conditions can contradict each other.

Assuming every "or" inequality produces two intervals

Overlapping "or" conditions can combine into one interval or even all real numbers.

Key Terms

Compound inequality — two inequalities combined to describe a solution set involving multiple conditions.

And inequality — compound inequality requiring both conditions to be satisfied.

Or inequality — compound inequality requiring at least one condition to be satisfied.

Double inequality — compact notation placing a variable between two inequality symbols, such as 2 < x ≤ 8.

Intersection — values shared by two solution sets; associated with "and."

Union — combined values belonging to either solution set; associated with "or."

Solution set — all values satisfying the compound inequality.

Boundary value — endpoint separating included and excluded values.

Interval notation — notation using parentheses and brackets to describe intervals of real numbers.

Union symbol (∪) — symbol used to combine separate intervals.

Open endpoint — boundary value not included in the solution.

Closed endpoint — boundary value included in the solution.

All real numbers — every real-number value, represented by (−∞, ∞).

Empty set (∅) — solution set containing no values.

Constraint — condition restricting the possible values of a variable.

Tolerance — acceptable range of variation around a desired measurement or operating value.

Key Takeaways

• Compound inequalities combine two mathematical conditions.
• And means both conditions must be true.
• Or means at least one condition must be true.
• "And" solutions are found by identifying the intersection or overlap.
• "Or" solutions are found by taking the union of acceptable values.
• Double inequalities such as 2 < x ≤ 8 represent "and" conditions.
• Operations on a double inequality must be performed on all three parts.
• Multiplying or dividing by a negative reverses every affected inequality symbol.
• Number lines provide a useful visual method for identifying compound solution sets.
• Open circles correspond to excluded endpoints.
• Closed circles correspond to included endpoints.
• Parentheses indicate excluded finite endpoints in interval notation.
• Brackets indicate included finite endpoints.
• Infinity always uses parentheses.
• Separate "or" intervals are joined with the union symbol ∪.
• An "and" inequality can sometimes have no solution.
• An "or" inequality can sometimes simplify to one interval or all real numbers.
• Compound inequalities are useful for expressing ranges, tolerances, safety limits, operating conditions, and restrictions.
• Testing boundary values helps determine whether endpoints should be included.
• A complete solution should clearly communicate the inequality, interval notation, and graphical representation.

Check Your Understanding

1. Explain the difference between an and compound inequality and an or compound inequality.
2. Write x > 3 and x < 9 as a double inequality and in interval notation.
3. Write x ≤ −4 or x > 6 in interval notation.
4. Solve: 2 < x + 4 ≤ 11. Give the solution as an inequality and in interval notation.
5. Solve: −12 ≤ 3x < 15. Describe the corresponding number-line graph.
6. Solve: −8 < −2x ≤ 10. Be careful when dividing by the negative coefficient.
7. Solve: 3x < −9 or 2x ≥ 8. Give the answer in interval notation.
8. Solve: 2x + 5 > 9 and 3x − 1 ≤ 14. Find the overlap of the two solution sets.
9. Solve: x > 8 and x < 3. Explain why the solution set is empty.
10. Simplify the solution: x ≥ 2 or x > 7. Explain why two intervals are unnecessary.
11. Determine whether x = 4 is a solution of −2 ≤ x < 4. Explain your answer using both the inequality and its interval notation.
12. A machine must operate at a temperature of at least 25°C but below 80°C. Write the compound inequality and interval notation.
13. A warning activates when pressure is below 30 kPa or above 90 kPa. Define a variable, write the compound inequality, and describe its number-line graph.
14. A manufactured component is acceptable when its length is between 74.95 mm and 75.05 mm, inclusive. Write the compound inequality and interval notation. Explain what the boundaries mean in the context of quality control.
15. A sensor is considered safe when −15 ≤ 3x + 6 ≤ 24. Solve the compound inequality, write the answer in interval notation, describe the number-line graph, and verify both boundary values.