Inequalities - Compound An inequality containing two inequalities or pair of inequalities combined

using the words "AND" or "OR" are called compound inequalities.

There are two types of compound inequalities. Those that are written as a

double inequality, and those connected by a word. Each are solved in slightly

different ways.

Example 1:

Solve the following compound inequality.

-3 < 4x + 5 < 25

Solution: When given a double inequality to solve, focus on solving for the x in the

middle.

Remember that whatever is done to the expression in the middle must also

be done to both the numbers on the outside of the inequalities.

-3 < 4x + 5 < 25

-8 < 4x < 20

-2 < x < 5

Therefore, the solution to the compound inequality is -2 < x < 5.

Example 2:

Solve the following compound inequality.

5x + 21 > 16 OR -5x - 10 < -20

Solution:

Solve each of the inequalities individually to get the correct answer.

5x + 21 > 16 OR -5x - 10 < -20

5x > -5 OR -5x < -10

x > -1 OR x > 2

Since x > -1 includes all the values of x such that x > 2, the solution is x > -

1.

Graph Inequalities A number line can be used to graph an inequality in one variable.

Read < as "less than". Read ≤ as "less than or equal to".

Read > as "greater than". Read ≥ as "greater than or equal to".

Example 1

Graph the inequality x > 1.

Notice the two different methods. In the top graph, an open dot is used to

show that 1 is not part of the solution, and in the bottom graph a parenthesis

is used.

Example 2

Graph the inequality x ≤ -2.

Again, there are two different methods. In the top graph, a closed dot is

used to show that -2 is part of the solution, and in the bottom graph a

bracket is used.

Absolute Value Inequalities To solve absolute value inequalities follow these steps.

1. Get the absolute value by itself on one side of the inequality.

2. Simplify within the absolute value bars, if possible, by combining like terms.

3. Separate the absolute value inequality into two cases, one for each sign.

4. Solve each of the two cases for the variable.

Example 1:

Solve the following absolute value inequality for x. |9x - (5x - 7)| ≥ 79 [Use

the four steps above.]

|9x - (5x - 7)| ≥ 79

|9x - 5x + 7| ≥ 79

|4x + 7| ≥ 79

Now, set up the two cases as explained below.

Case One: Write the inequality as it is without the absolute value bars.

Case Two: Drop the absolute value bars, flip the inequality symbol, and

change the sign of right side value.

4x + 7 ≥ 79

4x + 7 ≤ -79

4x ≥ 79 - 7

4x ≤ -79 - 7

4x ≥ 72

4x ≤ -86

x ≥ 18

OR

x ≤ -21.5

Look at the inequality symbol right before it was split into two cases. This symbol will decide whether the two answers will be AND or OR.

Greater Than (>) or Greater Than or Equal To (≥): Use OR. Less Than (<) or Less Than or Equal To (≤): Use AND.

Example 2:Solve the following absolute value inequality for x. |x -

12| - 4 < 21 [Use the four steps at the top.]

|x - 12| - 4 < 21

|x - 12| < 21 + 4

|x - 12| < 25

Now, set up the two cases as explained below.Case One: Write the

inequality as it is without the absolute value bars.

Case Two: Drop the absolute value bars, flip the inequality symbol, and

change the sign of right side value.

x - 12 < 25

x - 12 > -25

x < 25 + 12

x > -25 + 12

x < 37

AND

x > -13

When the answer has an AND, it is a conjunction and can be written in the

abbreviated form as shown below. -13 < x < 37

Remember to check your answers back into the original equation.

For x < 37, pick a number less than 37, for example 36.

For x > -13, pick a number greater than -13, for example -12.

|36 - 12| - 4 < 21

|-12 - 12| - 4 < 21

|24| - 4 < 21

|-24| - 4 < 21

24 - 4 < 21

24 - 4 < 21

20 < 21 true

20 < 21 true

Absolute Value Graphing

Less Than (<): Use an OPEN circle and Shade LEFT.

Greater Than (>): Use an OPEN circle and Shade

RIGHT.

Less Than or Equal To: Use a CLOSED circle and Shade

LEFT.

Greater Than or Equal

To:

Use a CLOSED circle and Shade

RIGHT.

Example 1: Graph the following absolute value inequality. |x| > 1

Split the absolute value into TWO cases and graph them on ONE number line.

x < -1 OR x > 1

Note: When the answer has OR, the arrows will shade OUT or away from the center.

Example 2: Graph the following absolute value inequality. |x| < 2

Split the absolute value into TWO cases and graph them on ONE number line.

x > -2 AND x < 2

Note: When the answer has AND, the arrows will shade IN or toward the center.

Example 3: Graph the following absolute value inequality. |x| < -3

Note: Absolute values are always POSITIVE, so they cannot be less than a negative

number. Therefore, the problem has NO solution.

Video Tutorials- 1. https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-and-

variables/cc-6th-inequalities/v/inequalities-on-a-number-line?v=dTwZ5N126gw

2. https://www.khanacademy.org/math/algebra/linear_inequalities/inequalities/v/multi-

step-inequalities-2?v=XOAn5z8mkvI

3. https://www.khanacademy.org/math/algebra/linear_inequalities/compound_absolute_

value_inequali/v/compund-inequalities?v=0YErxSShF0A

4. https://www.khanacademy.org/math/algebra/linear_inequalities/compound_absolute_

value_inequali/v/compund-inequalities-2?v=d2cnQ5ahHgE

5. https://www.khanacademy.org/math/algebra/linear_inequalities/compound_absolute_

value_inequali/v/compound-inequalities-3?v=cvB8b4AACyE