Solving Absolute Value Equations and Inequalities 1.7

The absolute value car

ride.

• What is absolute value?• What symbol is used to show absolute

value?• How do you transform an absolute value

inequality and know you have the signs going in the right direction?

Your house is located on: 0 Number Line Drive. If you are going to leave your house and go 3 miles, you could drive in either direction. If you traveled East 3 miles you would end up at 3 Number Line Drive and if you traveled West 3 miles you would end up at −3 Number Line Drive. Therefore, |x|=3; x=±3.

−10 −5 0 5 10

Absolute Value

The absolute value of a number x, written |x|, is the distance the number is from 0 on a number line. The absolute value of a number is always nonnegative.

Solving an absolute value equation.

The absolute value equation |ax+b| = c, where c>0, is equivalent to the compound statement ax + b = c or ax + b = −c.

Solving an Absolute Value Equation

Solve 952 x

x=7 or x=−2

Transformations of Absolute Value Inequalities

The inequality |ax+b| < c, where c > 0, means that ax+b is between −c and c. This is equivalent to −c < ax+b < c. (Less than goes with and)

The inequality |ax+b| > c, where c > 0, means that ax+b is beyond −c and c. This is equivalent to ax+b <−c or ax+b >c. (Greater than goes with or)

Solving with less than

Solve .1172 x

29 x

Solving with greater than

Solve 823 x

3102 xorx

A cereal manufacturer has a tolerance of 0.75 ounce for a box of cereal that is supposed to weigh 20 ounces. Write and solve an absolute value inequality that describes the acceptable weights for “20 ounce” boxes.

|Actual weight −ideal weight| < tolerance

75.2025.19 x

Quality controlYou are a quality control inspector at a bowling

pin company. A regulation pin must weigh between 50 ounces and 58 ounces, inclusive. Write an absolute value inequality describing the weights you should reject.

|Weight of pin − avg of extreme weights| > Tol.

454 x

• What is absolute value?• What symbol is used to show absolute

value?• How do you transform an absolute value

inequality and know you have the signs going in the right direction?

Check box on p. 51

Assignment p. 53, 18-54 even, 68, 74