1.6 Absolute Value Equations and Inequalities
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Transcript of 1.6 Absolute Value Equations and Inequalities
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1.6 ABSOLUTE VALUE EQUATIONS AND INEQUALITIES
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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
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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.
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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
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SOLVE EACH EQUATION. GRAPH THE SOLUTION.
2 1 5x
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SOLVE EACH EQUATION. GRAPH THE SOLUTION.
3 2 4x
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SOLVE EACH EQUATION. GRAPH THE SOLUTION.
3 2 1 8x
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SOLVE EACH EQUATION. GRAPH THE SOLUTION.
2 9 3 7x
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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.
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SOLVE AND CHECK FOR EXTRANEOUS SOLUTIONS.
2 8x x
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SOLVE AND CHECK FOR EXTRANEOUS SOLUTIONS.
3 2 4 5x x
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SOLVE AND CHECK FOR EXTRANEOUS SOLUTIONS.
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ABSOLUTE VALUE INEQUALITIES
An absolute value inequality is an inequality that has a variable inside the absolute value sign.
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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
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SOLVE AND GRAPH THE SOLUTION.
2 1 5x
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SOLVE AND GRAPH THE SOLUTION.
3 4 8x
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SOLVE AND GRAPH THE SOLUTION.
2 4 6x