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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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Other Common Inequality Phrases

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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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

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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.

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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

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.