Transcript 9 3.3 Absolute Value Equations and Inequalities Special Cases for

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