1 – 7 1 – 7 Solving Absolute Value Solving Absolute Value

Equations and InequalitiesEquations and Inequalities

Objective:

CA Standard 1: Students solve equations and inequalities involving absolute value.

The absolute value of x is the distance the number is from 0.

Can the absolute value of x ever be negative?

No

Solving an absolute value Solving an absolute value equationequation

The absolute value equation ax b c

where c > 0, is equivalent to the compound statement.

or ax b c ax b c

Solving an Absolute Value EquationSolving an Absolute Value Equation

Solve: 2 5 9x

Rewrite the absolute value equation as two linear equations and then solve each linear equation.

2 5 9x

2 5 9 or 2 5 9x x 2 14 or 2 4x x 7 or 2x x

An absolute value inequality such as:

2 4x can be solved by rewriting it as a compound

inequality:

4 2 4x

Transformation of Absolute Transformation of Absolute Value InequalitiesValue Inequalities

The inequality , where c>0, means that

is between and c. This is equivalent to

.

ax b c ax b

c

c ax b c

The inequality , where c>0,

means that is beyond -c and c. This is

equivalent to or .

ax b c

ax b

ax b c ax b c

Solving an inequality of the Solving an inequality of the form form ax + bax + b< c< c

Solve 2 7 11x 11 2 7 11x 18 2 2x

9 2x The solution is all real numbers

greater than –9 and less than 2. Graph the solution interval.

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

Solving an inequality of the Solving an inequality of the form form ax + b ax + b c c

Solve 3 2 8x This absolute value inequality is equivalent to

3 2 8 or 3 2 8x x 3 6x 3 10x

2x 10

3x

The solutions are all real numbers less than or equal to –2 and greater than or equal to 10/3.

Draw the graph of the solutions.

Are the dots open or closed?

Why?

Using Absolute Value in Using Absolute Value in Real lifeReal life

In manufacturing applications, the maximum deviation of a product from some ideal or average measurement is called a tolerance.

Writing a Model for ToleranceWriting a Model for Tolerance

A cereal manufacturer has a tolerance of 0.75 ounces 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.

Verbal Model:

Actual Weight – Ideal weight Tolerance

Labels:

Actual weight = x

Ideal weight = 20

Tolerance = .75

Algebraic Model: 20 .75x

.75 20 .75x

19.25 20.75x

Writing an Absolute Value ModelWriting an Absolute Value Model

You are a quality control inspector at a bowling pin company. A regulation pin

weighs between 50 and 58 ounces. Write an absolute value inequality describing the

weights you should reject.

Verbal Model:

Wt. of pin – Avg. wt. of extreme weights Tolerance

Labels: Weight of pin = w

Average weight of extreme weights

50 58 10854

2 2

Tolerance: 58 – 4 = 4

Algebraic Model: 54 4w

You should reject a bowling pin if its weight w satisfies

54 4w

HOMEWORK:HOMEWORK: