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.

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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.

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Inverse Operations

An inverse operation reverses another operation.

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

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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

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Solving Division Inequalities

Consider:

x/5 > 3

Multiply both sides by 5:

x > 15

Because 5 is positive, the inequality direction remains unchanged.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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Four Possible Symbol Reversals

When multiplying or dividing by a negative:

< becomes >

> becomes <

≤ becomes ≥

≥ becomes ≤

For example:

−2x ≤ 10

becomes:

x ≥ −5

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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.

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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.

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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.

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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.

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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 ≥

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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.

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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.

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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.

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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

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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.

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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.

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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

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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.

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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.

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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.

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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.

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Worked Example: Negative Multiplication

Solve:

−8x ≥ 32

Divide by −8.

Reverse ≥ to ≤:

x ≤ −4

Check:

−8(−4) ≥ 32

32 ≥ 32

True.

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Worked Example: Positive Division

Solve:

x/6 < 4

Multiply by 6:

x < 24

Check x = 18:

18/6 < 4

3 < 4

True.

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Worked Example: Negative Division

Solve:

x/−5 ≥ 6

Multiply by −5 and reverse:

x ≤ −30

Check:

−30/−5 ≥ 6

6 ≥ 6

True.

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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

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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.

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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.

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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.

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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.

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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.