Write out the compound inequalities.

1. x is at most -4 or at least 1

2. x is less than 7 and greater than or equal to 0

Solve the inequalities.

3. -11 < -3x – 2 < 1 4. -2x – 3 > 5 or 3x + 2 > -1

Warm-Up5.5

x < -4 or x > 1

0 < x < 7

-1 < x < 3

x < -4 or x > -1

Objectives:1. Solve absolute value equations2. Solve absolute value inequalities Vocabulary: absolute value

Solving Absolute Value Equations and Inequalities5.5

Absolute Value (|x|)the distance a number is away from the origin on a number line

The distance away from zero is less than 3 in both the positive and negative direction.

5.5

Simplify these expression:

|-5|

|7|

|-4 – 7|

|2(3 – 8)|

Absolute value – the distance a number is away from the origin on a number line

0 1 2 3 4 5 6 7 8 9 10 11-3 -2 -1-4-5-6-7-8-9-10-11

5

7

11

10

Solving Absolute Value Equations and Inequalities5.5

Solve this equation:

= 4x

x = 4 and x = -4

Solving Absolute Value Equations and Inequalities5.5

Solve this equation:

= 7x

x = 7 and x = -7

Solving Absolute Value Equations and Inequalities5.5

Solve this equation:

= -2x

No solution! An absolute value must always be

positive!

Solving Absolute Value Equations and Inequalities5.5

Solve this equation:

= 12

Solving Absolute Equations:

1. Isolate the absolute value.

2. Break it into 2 problems – one positive and the other negative.

3. Solve both equations.

4. Check both answers.

x + 4

x = 8 and x = -16

Solving Absolute Value Equations and Inequalities5.5

Solve this equation:

Solving Absolute Equations:

1. Isolate the absolute value.

2. Break it into 2 problems – one positive and the other negative.

3. Solve both equations.

4. Check both answers.

|3x – 3| = 6

x = 3 and x = -1

Solving Absolute Value Equations and Inequalities5.5

Solve this equation:

Solving Absolute Equations:

1. Isolate the absolute value.

2. Break it into 2 problems – one positive and the other negative.

3. Solve both equations.

4. Check both answers.

x = 6 and x = -2

|x – 2| + 4 = 8

Solve these equations:

Example 1 Solving an Absolute Value Equation

a.)|6x – 3|= 15 b.)|2x – 8|+ 2 = -22

Solving Absolute Equations:

1. Isolate the absolute value.

2. Break it into 2 problems – one positive and the other negative.

3. Solve both equations.

4. Check both answers.

x = 3 and x = -2

No solution; an absolute value can never give

a negative answer!

Solving an Absolute Value Inequality

Solving Absolute Inequalities:

1. Isolate the absolute value.

2. Set up the problem appropriately: “and” or “or.”

3. Solve the inequalities.

4. Check both answers.

Absolute inequality rules: |x|< c, then –c < x < c “and”|x|> c, then x < -c or x > c “or”

REMEMBER:<, <: less thAND>, >: greatORThe setup does not need absolute value symbols!

Solve and graph the inequality:

Watch Solving an Absolute Value Inequality

|x|< 5

Absolute inequality rules: |x|< c, then –c < x < c “and”|x|> c, then x < -c or x > c “or”

Solving Absolute Inequalities:

1. Isolate the absolute value.

2. Set up the problem appropriately: “and” or “or.”

3. Solve the inequalities.

4. Check both answers.

<, <: Less thAND>, >: GreatOR

-5 < x < 5

Solve and graph the inequalities:

Example 2 Solving an Absolute Value Inequality

a.) |4x – 6| < 22

b.)|3x – 9| > 18

Absolute inequality rules: |x|< c, then –c < x < c “and”|x|> c, then x < -c or x > c “or”

Solving Absolute Inequalities:

1. Isolate the absolute value.

2. Set up the problem appropriately: “and” or “or.”

3. Solve the inequalities.

4. Check both answers.

<, <: Less thAND>, >: GreatOR

-4 < x < 7

x < -3 or x > 9

Solve and graph the inequality:

Example 3 Solving an Absolute Value Inequality

|10 – 4x|≤ 2

Absolute inequality rules: |x|< c, then –c < x < c “and”|x|> c, then x < -c or x > c “or”

Solving Absolute Inequalities:

1. Isolate the absolute value.

2. Set up the problem appropriately: “and” or “or.”

3. Solve the inequalities.

4. Check both answers.

<, <: Less thAND>, >: GreatOR

3 > x > 2

2 < x < 3

Solve and graph the inequality:

Example 4 Solving an Absolute Value Inequality

|x + 2| – 5 ≥ 8

Absolute inequality rules: |x|< c, then –c < x < c “and”|x|> c, then x < -c or x > c “or”

Solving Absolute Inequalities:

1. Isolate the absolute value.

2. Set up the problem appropriately: “and” or “or.”

3. Solve the inequalities.

4. Check both answers.

<, <: Less thAND>, >: GreatOR

x < -15 or x > 11

Solve and graph the inequality:

Example 5 Solving an Absolute Value Inequality

|3x + 2| – 1 ≥ 10

Absolute inequality rules: |x|< c, then –c < x < c “and”|x|> c, then x < -c or x > c “or”

Solving Absolute Inequalities:

1. Isolate the absolute value.

2. Set up the problem appropriately: “and” or “or.”

3. Solve the inequalities.

4. Check both answers.

<, <: Less thAND>, >: GreatOR

x < -13/3 or x > 3

Solve and graph the inequality:

Example 6 Solving an Absolute Value Inequality

|2x + 5| – 1 < 6

Absolute inequality rules: |x|< c, then –c < x < c “and”|x|> c, then x < -c or x > c “or”

Solving Absolute Inequalities:

1. Isolate the absolute value.

2. Set up the problem appropriately: “and” or “or.”

3. Solve the inequalities.

4. Check both answers.

<, <: Less thAND>, >: GreatOR

-6 < x < 1

Example 8

RAINFALL The average annual rainfall in California for the last 100 years is 23 inches. However, the annual rainfall can differ by 10 inches from the 100 year average. What is the range of annual rainfall for California?

Apply Absolute Value Inequalities

Example 9

A. {x | 70 ≤ x ≤ 74}

B. {x | 68 ≤ x ≤ 72}

C. {x | 68 ≤ x ≤ 74}

D. {x | 69 ≤ x ≤ 75}

A thermostat inside Macy’s house keeps the temperature within 3 degrees of the set temperature point. If the thermostat is set at 72 degrees Fahrenheit, what is the range of temperatures in the house?

Connect Four Practice:

Algebra 1

• Each student will be responsible for working out every problem.

• If it is your turn and you get the problem correct, put your initials in any of the bottom squares (like dropping a chip in the Connect Four board).

• If it is your turn and you get the problem wrong, your partner will get a chance to “steal” your turn.

• Can’t agree on an answer? Raise your hand! Do not argue/shout/get worked up.

• Connect four in a row to win the game!Show your work for each problem on a separate sheet of paper! Number the problems. This will be

collected for points!

Assignment due tomorrow:

5.5 Homework Worksheet

Algebra 1