Download 2-8 Solving Absolute-Value Equations and Inequalities

2-8
Solving Absolute-Value
Equations and Inequalities
Who uses this?
Absolute value can be used to
represent the acceptable ranges for the
dimensions of baseball bats classified
by length or weight. (See Exercise 43.)
Objectives
Solve compound
inequalities.
Write and solve absolutevalue equations and
inequalities.
Vocabulary
disjunction
conjunction
absolute value
A compound statement is made up of
more than one equation or inequality.
A disjunction is a compound statement
that uses the word or.
n È { Ó
Dis- means “apart.”
Disjunctions have
two separate pieces.
Con- means
“together.”
Conjunctions
represent one piece.
ä
Ó
{
È
n
⎧
⎫
Disjunction: x ≤ -3 OR x > 2 Set builder notation: ⎨x|x ≤-3 x > 2⎬
⎩
⎭
A disjunction is true if and only if at least one of its parts is true.
A conjunction is a compound statement that uses the word and.
n È { Ó
ä
Ó
{
È
n
⎧
⎫
Conjunction: x ≥ -3 AND x < 2 Set builder notation: ⎨x|x ≥ -3 x < 2⎬
⎩ Conjunctions ⎭can be
A conjunction is true if and only if all of its parts are true.
written as a single statement as shown.
x ≥ -3 and x < 2 → -3 ≤ x < 2
EXAMPLE
1
Solving Compound Inequalities
Solve each compound inequality. Then graph the solution set.
A x + 3 ≤ 2 OR 3x > 9
Solve both inequalities for x.
x+3≤2
or
x ≤ -1
3x > 9
x>3
⎧
⎫
The solution set is all points that satisfy ⎨x | x ≤ -1 or x > 3⎬.
⎩
⎭
n È { Ó
ä
Ó
{
È
n
(-∞, -1] (3, ∞)
B -2x < 8 AND x - 3 ≤ 2
Solve both inequalities for x.
-2x < 8
and
x-3≤2
x > -4
x≤5
The solution set is the set of points that satisfy both x > -4 and x ≤ 5,
⎧
⎫
⎨ x | x - 4 < x ≤ 5⎬
⎩
⎭
n È { Ó
150
ä
Ó
{
È
n
(-4, 5]
Chapter 2 Linear Functions
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Solve each compound inequality. Then graph the solution set.
C x + 3 > 7 OR 3x ≥ 18
Solve both inequalities for x.
x+3>7
or
3x ≥ 18
x>4
x≥6
Because every point that satisfies x ≥ 6 also satisfies x > 4, the solution
⎧
⎫
set is ⎨ x | x > 4⎬.
⎩
⎭
{ Ó
ä
Ó
{
È
(4, ∞)
n
Solve each compound inequality. Then graph the solution set.
1a. x - 2 < 1 or 5x ≥ 30
1b. 2x ≥ -6 and -x > -4
1c. x - 5 < 12 or 6x ≤ 12
1d. -3x < -12 and x + 4 ≤ 12
Recall that the absolute value of a number x, written ⎪x⎥, is the distance from x
to zero on the number line. Because absolute value represents distance without
regard to direction, the absolute value of any real number is nonnegative.
Absolute Value
WORDS
NUMBERS
The absolute value of a real number
x, ⎪x⎥, is equal to its distance from
zero on a number line.
ALGEBRA
⎪5⎥ = 5
⎧ x if x ≥ 0
⎨
-x if x < 0
⎩
⎪x⎥ =
⎪-5⎥ = 5
Absolute-value equations and inequalities can be represented by compound
statements. Consider the equation ⎪x⎥ = 3.
Think: Greator
inequalities involving
> or ≥ symbols are
disjunctions.
Think: Less thand
inequalities involving
< or ≤ symbols are
conjunctions.
The solutions of ⎪x⎥ = 3 are the two
points that are 3 units from zero. The
solution is a disjunction: x = -3 or x = 3.
The solutions of ⎪x⎥ < 3 are the points
that are less than 3 units from zero. The
solution is a conjunction: -3 < x < 3.
The solutions of ⎪x⎥ > 3 are the points
that are more than 3 units from zero. The
solution is a disjunction:
x < -3 or x > 3.
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Ó
Î
{
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Absolute-Value Equations and Inequalities
For all real numbers x and all positive real numbers a:
⎪x⎥ = a
⎪x⎥ < a
x = -a OR x = a
x >-a AND x < a
⎪x⎥ > a
x < -a OR x > a
-a < x < a
Note: The symbol ≤ can replace <, and the rules still apply. The symbol ≥ can
replace >, and the rules still apply.
2- 8 Solving Absolute-Value Equations and Inequalities
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EXAMPLE
2
Solving Absolute-Value Equations
Solve each equation.
A ⎪x - 7⎥ = 5
This can be read as “the distance from x to 7 is 5.”
x - 7 = 5 or x - 7 = -5 Rewrite the absolute value as a disjunction.
x = 12 or x = 2
Add 7 to both sides of each equation.
B ⎪3x⎥ + 5 = 14
⎪3x⎥ = 9
Isolate the absolute-value expression.
3x = 9 or 3x = -9
x = 3 or x = -3
Rewrite the absolute value as a disjunction.
Divide both sides of each equation by 3.
Solve each equation.
2a. ⎪x + 9⎥ = 13
2b. ⎪6x⎥ - 8 = 22
You can solve absolute-value inequalities using the same methods that are used
to solve an absolute-value equation.
Solving an Absolute-value Inequality
1. Isolate the absolute-value expression, if necessary.
2. Rewrite the absolute-value expression as a compound inequality.
3. Solve each part of the compound inequality for x.
EXAMPLE
3
Solving Absolute-Value Inequalities with Disjunctions
Solve each inequality. Then graph the solution set.
A ⎪2x + 1⎥ > 5
In Example 3B, if you
recognize that
2x + 1 > 5 or 2x + 1 < -5
Rewrite the absolute value as a disjunction.
2x > 4 or 2x < -6
Subtract 1 from both sides of each inequality.
x > 2 or x < -3
⎫
⎧
⎨ x | x > 2 x < -3 ⎬
⎭
⎩
Divide both sides of each inequality by 2.
n È { Ó
⎪expression⎥ > -8
ä
Ó
{
È
(-∞, -3) (2, ∞)
n
To check, you can test a point in each of the three regions.
⎪2(-4) + 1⎥ > 5
⎪2(0) + 1⎥ > 5
⎪2(5) + 1⎥ > 5
⎪1⎥ > 5 ✘
⎪11⎥ > 5 ✔
⎪-7⎥ > 5 ✔
is always true,
you will know the
solution immediately.
B ⎪4x⎥ + 16 > 8
⎪4x⎥ > -8
Isolate the absolute-value expression.
4x > -8 or 4x < 8
Rewrite the absolute value as a disjunction.
x > -2 or x < 2
Divide both sides of each inequality by 4.
{
Ó
ä
Ó
{
È
n
(-∞, ∞)
The solution set is all real numbers, .
Solve each inequality. Then graph the solution set.
3a. ⎪4x - 8⎥ > 12
152
3b. ⎪3x⎥ + 36 > 12
Chapter 2 Linear Functions
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EXAMPLE
4
Solving Absolute-Value Inequalities with Conjunctions
Solve each inequality. Then graph the solution set.
A
⎪3x - 9⎥
_
≤ 12
2
⎪3x - 9⎥ ≤ 24
Multiply both sides by 2.
3x - 9 ≤ 24 and 3x - 9 ≥ -24
Rewrite the absolute value as a
conjunction.
3x ≤ 33 and
3x ≥ -15
x ≤ 11 and
x ≥ -5
Add 9 to both sides of each
inequality.
Divide both sides of each inequality
by 3.
⎧
⎫
The solution set is ⎨x |-5 ≤ x ≤ 11⎬.
⎩
⎭
È { Ó
In Example 4B, if you
recognize that
⎪expression⎥ ≤ -2
is never true, you will
know the solution
immediately.
ä
Ó
{
È
n £ä £Ó
B -4 ⎪x + 3⎥ ≥ 8
⎪x + 3⎥ ≤ -2
Divide both sides by -4, and reverse
the inequality symbol.
x + 3 ≤ -2 and x + 3 ≥ 2
Rewrite the absolute value as a
conjunction.
x ≤ -5 and x ≥ -1
Subtract 3 from both sides of each
inequality.
Because no real number satisfies both x ≤ -5 and x ≥ -1, there is no
solution. The solution set is ∅.
Solve each inequality. Then graph the solution set.
⎪x - 5⎥
4a. _ ≤ 4
2
4b. -2 ⎪x + 5⎥ > 10
THINK AND DISCUSS
1. Explain why the solution set to ⎪7x⎥ > -1 is all real numbers.
2. Explain why there is no solution to ⎪x + 3⎥ ≤ -2. Give another example
of an absolute-value equation that has no solution.
3. Write an absolute-value inequality to model “the distance between
x and 5 is greater than 10.”
4. GET ORGANIZED
Copy and complete the
graphic organizer.
Use the flowchart
to explain the
decisions and steps
needed to solve an
absolute-value
equation or inequality.
µÕ>̈œ˜ÊœÀ
ˆ˜iµÕ>ˆÌÞ¶
µÕ>̈œ˜
ˆÃÕ˜V̈œ˜ÊœÀ
Vœ˜Õ˜V̈œ˜¶
ˆÃÕ˜V̈œ˜
œ˜Õ˜V̈œ˜
2- 8 Solving Absolute-Value Equations and Inequalities
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