Lesson Objectives

- Demonstrate the ability to solve a linear inequality in one variable
- Learn how to solve a three-part linear inequality

## Solving a Three-Part Linear Inequality in One Variable

In some cases, we will see what is known as a "three-part" inequality. To solve a three-part inequality we isolate the variable in the middle.

Example 1: Solve each inequality, write the solution in interval notation, graph the interval

-11 ≤ 3x - 5 ≤ -2

We can still use our normal procedure, just remember our goal is to isolate the variable in the middle, not on one side.

Step 1) Simplify each part.

Each part here (-11), (3x - 5), and (-2) is already simplified.

Step 2) Isolate the variable term in the middle.

Since 5 is being subtracted away from x, we need to add 5 to each part:

-11 + 5 ≤ 3x - 5 + 5 ≤ -2 + 5

-6 ≤ 3x ≤ 3

Step 3) Isolate the variable in the middle.

We will divide each part by 3, the coefficient of x: $$\require{cancel}\frac{-6}{3}≤ \frac{3}{3}x ≤ \frac{3}{3}$$ $$\frac{-2\cancel{6}}{\cancel{3}}≤ \frac{1\cancel{3}}{\cancel{3}}x ≤ \frac{1\cancel{3}}{\cancel{3}}$$ $$-2 ≤ x ≤ 1$$ Interval Notation:

[-2, 1]

Graphing the Interval on the Number Line:

Example 1: Solve each inequality, write the solution in interval notation, graph the interval

-11 ≤ 3x - 5 ≤ -2

We can still use our normal procedure, just remember our goal is to isolate the variable in the middle, not on one side.

Step 1) Simplify each part.

Each part here (-11), (3x - 5), and (-2) is already simplified.

Step 2) Isolate the variable term in the middle.

Since 5 is being subtracted away from x, we need to add 5 to each part:

-11 + 5 ≤ 3x - 5 + 5 ≤ -2 + 5

-6 ≤ 3x ≤ 3

Step 3) Isolate the variable in the middle.

We will divide each part by 3, the coefficient of x: $$\require{cancel}\frac{-6}{3}≤ \frac{3}{3}x ≤ \frac{3}{3}$$ $$\frac{-2\cancel{6}}{\cancel{3}}≤ \frac{1\cancel{3}}{\cancel{3}}x ≤ \frac{1\cancel{3}}{\cancel{3}}$$ $$-2 ≤ x ≤ 1$$ Interval Notation:

[-2, 1]

Graphing the Interval on the Number Line:

#### Skills Check:

Example #1

Solve each inequality. $$5 < -7 - 2x < 13$$

Please choose the best answer.

A

$$-1 < x < 5$$

B

$$-10 < x < -6$$

C

$$-\frac{2}{3}< x < -\frac{1}{3}$$

D

$$-\frac{7}{5}< x < 2$$

E

$$x > -4$$

Example #2

Solve each inequality. $$-68 ≤ 10x - 8 < -58$$

Please choose the best answer.

A

$$-6 ≤ x < -5$$

B

$$-7 < x < 3$$

C

$$-1 < x < \frac{12}{5}$$

D

$$-5 ≤ x ≤ 5$$

E

$$-6 < x ≤ -5$$

Example #3

Solve each inequality. $$61 > 7x + 5 > 19$$

Please choose the best answer.

A

$$1 < x < 4$$

B

$$\frac{2}{5}< x < 13$$

C

$$-7 < x < 8$$

D

$$-8 < x < 17$$

E

$$2 < x < 8$$

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