Absolute Value Equations & Inequalities

EQ: How does solving an absolute value equation/inequality compare to solving a linear equation/inequality?

I am 3 “miles” from my house

|B| = 3B = 3 or B = -3

Review Absolute Value: the distance a number

is from zero (always positive)

Ex 1) |7| =

Ex 2) |-5| =

Ex 3) 5|2 – 4| + 2 =

Ex 1) Solve: |x| = 8

Ex 2) Solve: |x| = 25

Ex 3) Solve: |x| = -10

Absolute Value Equations

Have two solutions because the absolute

value of a positive number is the same

as the absolute value of a negative

number.

Solving Absolute Value Equations

Step 1: Get “bars” alone on one side

Step 2: Re-write as two equations; flip the

signs on the RIGHT side (drop the bars)

Step 3: Solve both equations

Step 4: Check for an extraneous solution!!

Extraneous Solution A value you get after correctly solving

the problem that does not actually satisfy the equation.

Ex 4)

3|x + 2| - 7 = 14

3|x + 2| = 21

|x + 2| = 7

|x + 2| = 7

x + 2 = 7 x + 2 = -7 x = 5 or x = -9

Check Your Answers:

x = 5 x = -9

3|x + 2| - 7 = 14 3|x + 2| - 7 = 14

3|5 + 2| - 7 = 14 3|-9 + 2| - 7 = 14

3|7| - 7 = 14 3|-7| - 7 = 14

14 = 14 14 = 14

Ex 5)

|3x + 2| = 4x + 5

3x + 2 = 4x + 5

3x + 2 = -4x – 5

3x + 2 = 4x + 5

3x + 2 = -4x – 5

|3x + 2| = 4x + 5

Check Your Solutions

Ex 6)

|x – 4| + 7 = 2

|x – 4| = -5

NO SOLUTION

Solving by Graphing

Step 1: Enter the left & right side into Y1 & Y2

PRESS → NUM #1 abs(

Step 2: Find the first intersection

Step 3: Find the second intersection if there is one

MATH

2nd TRACE #5

3|x + 2| -7 = 14

|3x + 2| = 4x + 5

Practice

1) Solve: |5x| + 10 = 55

2) Solve: |x – 3| = 10

3) Solve: 2|y + 6| = 8

4) Solve: |a – 5| + 3 = 2

5) Solve: |4x + 9| = 5x + 18

Thank about it… If |x| < 5 what are some possible values

of x?

If |x| > 5 what are some possible values of x?

Solving Absolute Value Inequalities

  Since the inequalities are absolute value, there are still going to be two solutions

When writing the second equation, be sure to flip the inequality sign.

Also, when dividing by a negative the inequality sign flips.

Ex 1)

Ex 2)

You Try! Solve | 2x + 3 | < 6

Ex 3) Solve 423 xx

You Try! Solve |2x-5|>x+1.

Quick Write: Think of some objects or situations that need

to be within a certain “value” – if you go too much over it would be a bad thing, and if you go too much under it would also be a bad thing:

Tolerance

There are strict height requirements to be a “Rockette”

You must be between 66 inches 70.5 inches

Perfect Amount

LeastAllowed

MostAllowed

Tolerance Tolerance

Tolerance: The difference between a desired

measurement and its maximum and minimum allowable values.

Ex 1) The doctor says that you need to stay between 125 and 135 lbs to be at a healthy weight.

Min: _______ Perfect: _______ Max: ________

Tolerance: __________ 

Example 2) Workers at a hardware store take their

morning break no earlier than 10 am and no later than noon. Let c represent the time the workers take their break. Write an absolute value inequality to represent the situation.

Exit Ticket: How is solving an absolute value inequality

different than solving an absolute value equation?

What is the difference between the solutions of an equation and the solutions of an inequality?