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. We will perform the same action to each part until we accomplish our goal of:

some number < x < some number

Let's look at a few examples.

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

-11 ≤ 3x - 5 ≤ -2

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

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 2: Solve each inequality, write in interval notation, graph.

-7 ≤ x - 1 ≤ 9

To isolate x in the middle, let's add 1 to each part:

-7 + 1 ≤ x - 1 + 1 ≤ 9 + 1

-6 ≤ x ≤ 10

Interval Notation:

[-6, 10]

Graphing the Interval: Example 3: Solve each inequality, write in interval notation, graph.

-90 < -9x ≤ -27

To isolate x in the middle, let's divide each part by (-9). Remember, this means we have to flip each inequality symbol.

-90/-9 > -9/-9 x ≥ -27/-9

10 > x ≥ 3

Write this in the direction of the number line:

3 ≤ x < 10

Interval Notation:

[3, 10)

Graphing the Interval:

some number < x < some number

Let's look at a few examples.

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

-11 ≤ 3x - 5 ≤ -2

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

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 2: Solve each inequality, write in interval notation, graph.

-7 ≤ x - 1 ≤ 9

To isolate x in the middle, let's add 1 to each part:

-7 + 1 ≤ x - 1 + 1 ≤ 9 + 1

-6 ≤ x ≤ 10

Interval Notation:

[-6, 10]

Graphing the Interval: Example 3: Solve each inequality, write in interval notation, graph.

-90 < -9x ≤ -27

To isolate x in the middle, let's divide each part by (-9). Remember, this means we have to flip each inequality symbol.

-90/-9 > -9/-9 x ≥ -27/-9

10 > x ≥ 3

Write this in the direction of the number line:

3 ≤ x < 10

Interval Notation:

[3, 10)

Graphing the Interval:

#### 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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