1.7: Solve Absolute Value Equations and 1.7: Solve Absolute Value Equations and InequalitiesInequalities

Objectives:

1.To solve and graph absolute value equations and inequalities

Exercise 1Exercise 1

Kenny digs a hole in his backyard while his parents are still at work. He neatly piles the dirt from the hole on the concrete patio. When Kenny’s parents return home, they demand that Kenny explain why he destroyed their well-manicured backyard oasis. Kenny says that the hole and the dirt from the whole demonstrate absolute value. What is Kenny talking about?

Absolute ValueAbsolute Value

The absolute value absolute value of a number is its distance from zero on a real number line.

• The absolute value is always positive.

Absolute Value EquationsAbsolute Value Equations

Solving Absolute Value Solving Absolute Value EquationsEquations

To solve |ax + b| = c,

1.Write TWO equations:– ax + b = c

– ax + b = -c

• Solve each equation.

• Check each solution in the original equation.

Exercise 2 Exercise 2

Solve each absolute value equation.

1.|x| = 15

2.|2x – 9| = 15.

Exercise 3Exercise 3

Solve.

1.|4x + 12| = 28

2.|4x + 10| = 6x

Extraneous SolutionsExtraneous Solutions

In the course of solving an absolute value equation, one of the solutions may not actually satisfy the original equation. This an extraneous solutionextraneous solution. Get rid of it; it’s no good!

Try in your notebooks

• Page 55 3-6, 9-15, 21-24, 34-36

Absolute Value InequalitiesAbsolute Value Inequalities

Follow me here:

• |x| = 5 means the distance from zero equals 5.

• |x| ≤ 5 means the distance from zero is less than or equal to five.

-5 ≤ x ≤ 5

-5 50

-5 50

Absolute Value InequalitiesAbsolute Value Inequalities

Follow me here:

• |x| = 5 means the distance from zero equals 5.

• |x| ≥ 5 means the distance from zero is greater than or equal to five.

x ≤ -5 or x ≥ 5

-5 50

-5 50

Absolute Value InequalitiesAbsolute Value Inequalities

If the general methods from the previous slides are incomprehensible, you could just memorize these.

Exercise 4Exercise 4

Solve the inequality. Then graph the solution.

1.|x + 2| < 6

2.|2x + 1| ≤ 9

3.|7 – x| ≤ 4

Exercise 5Exercise 5

Solve the inequality. Then graph the solution.

1.|x + 4| ≥ 6

2.|2x – 7| > 1

3.|3x + 5| ≥ 10

Exercise 6Exercise 6

A professional baseball should weigh 5.125 ounces, with a tolerance of 0.125 ounces. (Tolerance is the maximum deviation from the ideal measurement.) Write and solve an absolute value inequality that describes the acceptable weights of a baseball.

Solution

• First write a verbal model of the equation:– │actual weight- ideal weight │≤ tolerance

• Convert verbal model to an equation and solve:– │w-5.125 │≤ 0.125– -0.125≤ w-5.125≤ 0.125– 5 ≤ w ≤ 5.25

AssignmentAssignment

• P. 55: 40,43-54,74, 76a-b