copyright © 2010 pearson education, inc. all rights reserved sec 3.3 - 1 3.3 absolute value...
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Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 3.3 - 1
3.3
Absolute Value Equations
and Inequalities
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.3 - 2
3.3 Absolute Value Equations and InequalitiesSummary:
Solving Absolute Value Equations and Inequalities
1. To solve |ax + b| = k, solve the following compound equation.
Let k be a positive real number, and p and q be real numbers.
ax + b = k or ax + b = –k.
The solution set is usually of the form {p, q}, which includes two
numbers.
p q
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.3 - 3
3.3 Absolute Value Equations and InequalitiesSummary:
Solving Absolute Value Equations and Inequalities
2. To solve |ax + b| > k, solve the following compound inequality.
Let k be a positive real number, and p and q be real numbers.
ax + b > k or ax + b < –k.
The solution set is of the form (-∞, p) U (q, ∞), which consists of two
separate intervals.
p q
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.3 - 4
3.3 Absolute Value Equations and InequalitiesSummary:
Solving Absolute Value Equations and Inequalities
3. To solve |ax + b| < k, solve the three-part inequality
Let k be a positive real number, and p and q be real numbers.
–k < ax + b < k
The solution set is of the form (p, q), a single interval.
p q
3.3 Absolute Value Equations and Inequalities
EXAMPLE 1 Solving an Absolute Value Equation
Solve |2x + 3| = 5.
3.3 Absolute Value Equations and Inequalities
EXAMPLE 2 Solving an Absolute Value Inequality with >
Solve |2x + 3| > 5.
3.3 Absolute Value Equations and Inequalities
EXAMPLE 3 Solving an Absolute Value Inequality with <
Solve |2x + 3| < 5.
3.3 Absolute Value Equations and Inequalities
EXAMPLE 4 Solving an Absolute Value Equation That
Requires Rewriting
Solve the equation |x – 7| + 6 = 9.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.3 - 9
3.3 Absolute Value Equations and Inequalities
Special Cases for Absolute Value
Special Cases for Absolute Value
1. The absolute value of an expression can never be negative: |a| ≥ 0
for all real numbers a.
2. The absolute value of an expression equals 0 only when the expression is equal to 0.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.3 - 10
3.3 Absolute Value Equations and Inequalities
EXAMPLE 6 Solving Special Cases of Absolute Value
Equations
Solve each equation.
See Case 1 in the preceding slide. Since the absolute value of an expression can never be negative, there are no solutions for this equation.The solution set is Ø.
(a) |2n + 3| = –7
See Case 2 in the preceding slide. The absolute value of the expres-sion 6w – 1 will equal 0 only if
6w – 1 = 0.
(b) |6w – 1| = 0
The solution of this equation is . Thus, the solution set of the original
equation is { }, with just one element. Check by substitution.
16
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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.3 - 11
3.3 Absolute Value Equations and Inequalities
EXAMPLE 7 Solving Special Cases of Absolute Value
Inequalities
Solve each inequality.
The absolute value of a number is always greater than or equal to 0.Thus, |x| ≥ –2 is true for all real numbers. The solution set is (–∞, ∞).
(a) |x| ≥ –2
Add 1 to each side to get the absolute value expression alone on oneside.
|x + 5| < –7
(b) |x + 5| – 1 < –8
There is no number whose absolute value is less than –7, so this inequalityhas no solution. The solution set is Ø.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.3 - 12
3.3 Absolute Value Equations and Inequalities
EXAMPLE 7 Solving Special Cases of Absolute Value
Inequalities
Solve each inequality.
Subtracting 2 from each side gives
|x – 9| ≤ 0
(c) |x – 9| + 2 ≤ 2
The value of |x – 9| will never be less than 0. However, |x – 9| will equal 0when x = 9. Therefore, the solution set is {9}.