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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