1.6 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES

ABSOLUTE VALUE

The absolute value of a real number x, written |x|, is its distance from zero on the number line Example: |5| = 5

|-5| = 5

ABSOLUTE VALUES EQUATION

An absolute value equation is an equation that has a variable inside the absolute value sign Absolute value equations can have two answers

because opposites have the same absolute value.

SOLVING ABSOLUTE VALUE EQUATIONS To solve an absolute value equation:

1. Isolate the absolute value2. remove the absolute value signs and set up as

shown:

or

x k

x k x k

SOLVE EACH EQUATION. GRAPH THE SOLUTION.

2 1 5x

SOLVE EACH EQUATION. GRAPH THE SOLUTION.

3 2 4x

SOLVE EACH EQUATION. GRAPH THE SOLUTION.

3 2 1 8x

SOLVE EACH EQUATION. GRAPH THE SOLUTION.

2 9 3 7x

EXTRANEOUS SOLUTIONS An extraneous solution is a solution derived

from an original equation that is not a solution of the original equation.

Remember that the absolute value measures the distance from zero on a number line. Distance can never be negative. Therefore, we must check our answers when working with absolute values.

SOLVE AND CHECK FOR EXTRANEOUS SOLUTIONS.

2 8x x

SOLVE AND CHECK FOR EXTRANEOUS SOLUTIONS.

3 2 4 5x x

SOLVE AND CHECK FOR EXTRANEOUS SOLUTIONS.

ABSOLUTE VALUE INEQUALITIES

An absolute value inequality is an inequality that has a variable inside the absolute value sign.

SOLVING ABSOLUTE VALUE INEQUALITIES

Write the absolute value inequality as a compound inequality without absolute value symbols

Compound :

Compound :

x aand

x a

x aor

x a

SOLVE AND GRAPH THE SOLUTION.

2 1 5x

SOLVE AND GRAPH THE SOLUTION.

3 4 8x

SOLVE AND GRAPH THE SOLUTION.

2 4 6x