Training Linear Equations and Inequalities Solving Linear Inequalities
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Solving Linear Inequalities

20 min Linear Equations and Inequalities

Linear Inequalities

Inequalities are close cousins of equations. Instead of asking "what value makes these two expressions equal," an inequality asks "what values make one expression less than (or greater than) the other?" The result is not a single number but an entire range of solutions.

Solving linear inequalities uses the same inverse-operation techniques as equations, with one crucial difference: multiplying or dividing both sides by a negative number reverses the inequality sign.

This lesson covers solving, graphing, and writing the solution sets of linear inequalities using both interval and set-builder notation.

An inequality uses $<$, $>$, $\le$, or $\ge$ instead of $=$.

Key Rule

Solve exactly like equations, except: when you multiply or divide by a negative number, reverse the inequality sign.

Example 1

$3x + 5 > 14$

$3x > 9 \;\Rightarrow\; x > 3$. Graph: open circle at 3, arrow right.

Example 2

$-2x + 7 \le 15$

$-2x \le 8 \;\Rightarrow\; x \ge -4$ (reversed!). Graph: closed circle at $-4$, arrow right.

Example 3

$4 - 3x < 10$

$-3x < 6 \;\Rightarrow\; x > -2$ (reversed!).

Compound Inequalities

Example 4

$-3 < 2x + 1 \le 7$

Subtract 1: $-4 < 2x \le 6$. Divide by 2: $-2 < x \le 3$.

Example 5

$3x - 1 < 5$ or $2x + 3 > 11$

$x < 2$ or $x > 4$.

Practice Problems

1. $5x - 3 > 12$
2. $-4x + 1 \ge 9$
3. $2(x - 3) \le 8$
4. $-1 \le 3x + 2 < 14$
5. $7 - x > 3$
6. $\dfrac{x}{3} + 2 < 5$
7. $-5 < 2x - 1 \le 9$
8. $4x + 7 > 2x - 3$
9. $-3(x - 4) \ge 6$
10. $2x + 1 \le 5$ or $3x - 2 > 10$
11. $8 - 5x < -7$
12. $\dfrac{2x - 1}{3} \ge 3$
Show Answer Key

1. $x > 3$

2. $x \le -2$

3. $x \le 7$

4. $-1 \le x < 4$

5. $x < 4$

6. $x < 9$

7. $-2 < x \le 5$

8. $x > -5$

9. $x \le 2$

10. $x \le 2$ or $x > 4$

11. $x > 3$

12. $x \ge 5$

Multi-Step Equations Absolute Value Equations and Inequalities