Absolute Value Equations

Objective: Solve equations with an absolute value in them; identify extraneous solutions

Essential Question: How do absolute value bars affect the possible solutions of an equation?

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 =

Consider the following problem…

|x + 4| = 7

Solving Absolute Value Equations

Step 1: Get “bars” alone on one side

Step 2: Split up into a positive & negative equation (drop the bars)

Step 3: Solve both equations

Step 4: Check your solution!!! *One of them might be

extraneous

Problem #1

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

Problem #2

|3x + 2| = 4x + 5

3x + 2 = 4x + 5

3x + 2 = -4x – 5

3x + 2 = 4x + 5

3x + 2 = -4x – 5

Extraneous Solutions – solution that you find that is NOT actually a solution

It’s a FAKE! |3x + 2| = 4x + 5

Problem #3

|x – 4| + 7 = 2

|x – 4| = -5

NO SOLUTION

NO SOLUTION

If the equivalent value is

negative BEFORE YOU DROP

THE BARS, there is no solution

Problem #4

|x – 4| < 10

x – 4 < 10 x – 4 > -10

FLIP SIGN

Practice

Pg.55 #1 – 11 ODD

Pg.56 #17 – 23 ODD

You will be turning this in

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

Problem #1 3|x + 2| -7 = 14

Problem #2 |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

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

Problem #1In a car racing, a car must meet specific dimensions to enter a race. What absolute value inequality describes the heights of the model of race cars with a desirable height of 52 inches, a greatest allowable height of 53 inches, and a least allowable height of 51 inches?

Greatest =Least =Tolerance =Ideal Amount =Actual Amount =

Problem #2A manufacturer has a 0.6 oz tolerance for a bottle of salad dressing advertised as 16oz. Write and solve an absolute value inequality that describes the acceptable volumes for 16 oz.

Greatest =Least =Tolerance =Ideal Amount =Actual Amount =

Problem #3Suppose you used an oven thermometer while baking and discovered that the oven temperature varied between + 5 and -5 degrees from the setting. If your oven is set to 350*, let t be the actual temperature. Write an absolute value inequality to represent the situation.

Greatest =Least =Tolerance =Ideal Amount =Actual Amount =

Problem #4A distributor has a tolerance of 0.36 lb for a bag of potting soil advertised as 9.6lb. Write and solve an absolute value inequality that describes acceptable weights for a bag.

Greatest =Least =Tolerance =Ideal Amount =Actual Amount =

Problem #5In a newspaper poll taken before an election, 42% of the people favor the incumbent mayor. The margin of error for the actual percentage p is less than 4%. Find an absolute value inequality that represents this situation.

Greatest =Least =Tolerance =Ideal Amount =Actual Amount =

Problem #6In a wood shop, you have to drill a hole that is two inches deep into a wood panel. The tolerance for drilling a hole is 0.125 inches. What is the shallowest hole allowed?

Greatest =Least =Tolerance =Ideal Amount =Actual Amount =

Problem #7A survey reveals that 78% of people in North Carolina favor a particular law being passed. The margin of error for the actual percentage p is 5%. Write an inequality to model this situation

Greatest =Least =Tolerance =Ideal Amount =Actual Amount =

Problem #8The normal thickness of a metal structure is 6.5 cm. It expands to 6.54 centimeters when heated and shrinks to 6.46 cm when cooled down. What is the maximum amount in cm that the thickness of the structure can deviate from its normal thickness?

Greatest =Least =Tolerance =Ideal Amount =Actual Amount =