How to Solve One-Variable Linear Inequalities

By Grade 7 at Think Academy (Grade 8 in school), students work with one-variable linear inequalities. The challenge is not the arithmetic itself but remembering the rules—many forget to flip the inequality sign when multiplying or dividing by negatives, or mix up the direction of shading on number lines. This guide breaks down the steps so those mistakes don’t block progress.

What Are Inequalities?

An inequality is like an equation, but instead of showing two sides are equal, it shows that one side is bigger, smaller, or possibly equal to the other.

Inequalities describe a range of possible answers, not just one solution.

What Is the Solution to an Inequality?

The solution to an inequality is all the values of the variable that make the statement true.

For example:

𝑥 > 5

means any number greater than 5 works.

Note the position of the variable and the number:

This makes our answer easier to read and avoids confusion.

For example, if our work leads to:

5 < 𝑥,

flip and rewrite it as:

𝑥 > 5.

The inequality sign retains its original meaning: the larger side continues to point toward the larger quantity, and the smaller side continues to point toward the smaller quantity.

How to Solve One-Variable Linear Inequalities

Solving inequalities is very similar to solving equations. The difference is that we must be careful about the direction of the inequality sign.

Addition

Adding the same number to both sides does not change the inequality sign.

𝑥 – 3 > 7

Add 3 to both sides:

𝑥 – 3 + 3 > 7 + 3

𝑥 > 10

Subtraction

Subtracting the same number from both sides also does not change the inequality sign.

𝑥 + 4 ≤ 12

Subtract 4 from both sides:

𝑥 + 4 − 4 ≤ 12 − 4
𝑥 ≤ 8

Multiplication

• If we multiply both sides by a positive number, the inequality direction stays the same.

\[\frac{1}{2}x < 5\]

Multiply both sides by 2:

\[\frac{1}{2}x \cdot 2 < 5 \cdot 2\]

𝑥 < 10

• If we multiply both sides by a negative number, reverse the inequality sign.

\[-\frac{1}{3}x > 7\]

Multiply both sides by -3:

\[\left(-\frac{1}{3}x\right) \cdot (-3) < (7) \cdot (-3)\]

x < -21

Division

The rules are the same as multiplication:

• Divide by a positive number → sign stays the same.

2𝑥 ≥ 10

Divide both sides by 2:

2𝑥 ÷ 2 ≥ 10 ÷ 2
𝑥 ≥ 5

• Divide by a negative number → sign flips.

−3𝑥 < −6

Divide both sides by -3:

(−3𝑥) ÷ (−3) > (−6) ÷ (−3)
𝑥 > 2

Example Problems: Solving One-Variable Linear Inequalities

Example 1

Solve the inequality and graph the solution on the number line.

5𝑥 – 7 > 8

Solution:

Add 7 to both sides:

5𝑥 – 7 + 7 > 8 + 7

5𝑥 > 15

Divide both sides by 5:

5𝑥 ÷ 5 > 15 ÷ 5
𝑥 > 3

Example 2

Solve the inequality and graph the solution on the number line.

−2𝑥 + 4 ≤ 10

Solution:

Subtract 4 from both sides:

−2𝑥 + 4 − 4 ≤ 10 − 4
−2𝑥 ≤ 6

Divide both sides by -2 and flip sign:

(−2𝑥) ÷ (−2) ≥ 6 ÷ (−2)
𝑥 ≥ −3

Summary: Solving One-Variable Linear Inequalities

When solving inequalities:

• Adding or subtracting any number → inequality sign stays the same.
• Multiplying or dividing by a positive number → inequality sign stays the same.
• Multiplying or dividing by a negative number → inequality sign flips.

Want more printable practice?

Additional Math Topics for Grade 8 – with Free Worksheets

• How to Solve Equations with Variables on Both Sides in 3 Simple Steps
• How to Solve Equations with Parentheses in 4 Simple Steps
• How to Solve Equations with Fractions and Decimals in 3 Simple Steps

About Think Academy

Think Academy, a leading K–12 math education provider wholly owned by TAL Education Group, is dedicated to helping students build strong mathematical foundations and critical thinking. Our structured curriculum provides multiple course levels designed to accommodate students with diverse academic goals and proficiency levels, ensuring targeted and effective learning experiences. Supported by advanced teaching methods, expert instructors, and innovative AI technology, Think Academy consistently demonstrates excellence, trustworthiness, and proven expertise in mathematics education.

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