Lesson Objectives

- Demonstrate an understanding of interval notation
- Demonstrate an understanding of how to graph an interval on a number line
- Demonstrate an understanding of the addition property of inequality
- Demonstrate an understanding of the multiplication property of inequality
- Learn how to solve any Linear Inequality in One Variable
- Learn how to solve a Three-Part Linear Inequality in One Variable

## How to Solve a Linear Inequality in One Variable

In the last lesson, we learned the basics of how to solve a linear inequality in one variable. In that lesson, we learned about interval notation, how to graph an interval on the number line, the addition property of inequality, and the multiplication property of inequality. It is very important to understand these concepts before moving forward. In this lesson, we will focus on a general procedure to solve any linear inequality in one variable.

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

31 + 8x ≥ 5(3x - 5)

1) Simplify each side

31 + 8x ≥ 15x - 25

2) Isolate the variable term on one side

31 - 31 + 8x ≥ 15x - 25 - 31

8x ≥ 15x - 56

8x - 15x ≥ 15x - 15x - 56

-7x ≥ -56

3) Isolate the variable

-7/-7 x ≥ -56/-7

x ≤ 8

Remember, if we divide by a negative, we flip the direction of the inequality symbol.

Interval Notation:

(-∞,8]

Graphing the Interval: Example 2: Solve each inequality, write in interval notation, graph.

7(x - 8) - 8x > -8 + 7x

1) Simplify each side

7x - 56 - 8x > -8 + 7x

-x - 56 > -8 + 7x

2) Isolate the variable term on one side

-x - 56 + 56 > -8 + 56 + 7x

-x > 48 + 7x

-x - 7x > 48 + 7x - 7x

-8x > 48

3) Isolate the variable

-8/-8 x > 48/-8

x < -6

Remember, if we divide by a negative, we flip the direction of the inequality symbol.

Interval Notation:

(-∞,-6)

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

-9(1 + 8x) - 4x ≥ - 10(9x + 9) + 11

1) Simplify each side

-9 - 72x - 4x ≥ -90x - 90 + 11

-9 - 76x ≥ -90x - 79

2) Isolate the variable term on one side

-9 + 9 - 76x ≥ -90x - 79 + 9

-76x ≥ -90x - 70

-76x + 90x ≥ -90x + 90x - 70

14x ≥ -70

3) Isolate the variable

14/14 x ≥ -70/14

x ≥ -5

Interval Notation:

[-5,∞)

Graphing the Interval:

some number < x < some number

Let's look at a few examples.

Example 4: 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 5: 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:

### Solving a Linear Inequality in One Variable

- Simplify each side completely
- Just like with equations, we can clear decimals or fractions

- Isolate the variable term on one side of the inequality
- We perform this action using the addition property of inequality

- Isolate the variable
- We perform this action using the multiplication property of inequality

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

31 + 8x ≥ 5(3x - 5)

1) Simplify each side

31 + 8x ≥ 15x - 25

2) Isolate the variable term on one side

31 - 31 + 8x ≥ 15x - 25 - 31

8x ≥ 15x - 56

8x - 15x ≥ 15x - 15x - 56

-7x ≥ -56

3) Isolate the variable

-7/-7 x ≥ -56/-7

x ≤ 8

Remember, if we divide by a negative, we flip the direction of the inequality symbol.

Interval Notation:

(-∞,8]

Graphing the Interval: Example 2: Solve each inequality, write in interval notation, graph.

7(x - 8) - 8x > -8 + 7x

1) Simplify each side

7x - 56 - 8x > -8 + 7x

-x - 56 > -8 + 7x

2) Isolate the variable term on one side

-x - 56 + 56 > -8 + 56 + 7x

-x > 48 + 7x

-x - 7x > 48 + 7x - 7x

-8x > 48

3) Isolate the variable

-8/-8 x > 48/-8

x < -6

Remember, if we divide by a negative, we flip the direction of the inequality symbol.

Interval Notation:

(-∞,-6)

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

-9(1 + 8x) - 4x ≥ - 10(9x + 9) + 11

1) Simplify each side

-9 - 72x - 4x ≥ -90x - 90 + 11

-9 - 76x ≥ -90x - 79

2) Isolate the variable term on one side

-9 + 9 - 76x ≥ -90x - 79 + 9

-76x ≥ -90x - 70

-76x + 90x ≥ -90x + 90x - 70

14x ≥ -70

3) Isolate the variable

14/14 x ≥ -70/14

x ≥ -5

Interval Notation:

[-5,∞)

Graphing the Interval:

### Solving a Three-Part Inequality

We will also come across three-part inequalities. To solve these, we just follow a similar set of steps. Our goal is to 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 4: 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 5: 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 the inequality, write the answer in interval notation. $$8(8 + 4x) + 2 ≥ 98$$

Please choose the best answer.

A

$$[4, \infty)$$

B

$$[-1, \infty)$$

C

$$[-4, \infty)$$

D

$$[1, \infty)$$

E

$$(-\infty, -4]$$

Example #2

Solve the inequality, write the answer in interval notation. $$6 < 5(1 + 4x) - 7(-4x - 7)$$

Please choose the best answer.

A

$$(-1, \infty)$$

B

$$(-\infty, -1)$$

C

$$(-32, \infty)$$

D

$$\left(\frac{5}{3}, \infty\right)$$

E

$$(-18, \infty)$$

Example #3

Solve the inequality, write the answer in interval notation. $$41 - 9x ≤ 2(x + 4)$$

Please choose the best answer.

A

$$[3, \infty)$$

B

$$[-3, \infty)$$

C

$$(-\infty, -3]$$

D

$$(4, \infty)$$

E

$$\left[-\frac{8}{17}, \infty\right)$$

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