Download 7.6 Lecture Notes

Transitioning to
Intermediate Algebra
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7
Section
7.6
Solving Equations in One Variable
Containing an Absolute Value Expression
Copyright © Cengage Learning. All rights reserved.
Objectives
1 Solve an equation containing a single absolute
value expression.
2 Solve an equation where two absolute value
expressions are equal.
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Solving Equations in One Variable Containing an Absolute
Value Expression
In this section, we will review the definition of absolute
value and show how to solve equations that contain
absolute values.
We know the following definition of the absolute value of x.
Absolute Value
If x  0, then |x| = x.
If x < 0, then |x| = –x.
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Solving Equations in One Variable Containing an Absolute
Value Expression
This definition associates a nonnegative real number with
any real number.
• If x  0, then x is its own absolute value.
• If x < 0, then –x (which is positive) is the absolute value.
Either way, |x| is positive or 0:
|x|  0
for all real numbers x
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1. Solve an equation containing a
single absolute value expression
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Solve an equation containing a single absolute value expression
In the equation |x| = 5, x can be either 5 or –5, because
|5| = 5
and |–5| = 5
Thus, if |x| = 5, then x = 5 or x = –5. In general, the
following is true.
Absolute Value Equations
If k > 0, then
|x| = k
is equivalent to
x = k or x = –k
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Solve an equation containing a single absolute value expression
The absolute value of x can be interpreted as the distance
on the number line from a point to the origin.
The solutions of |x| = k are represented by the two points
that lie exactly k units from the origin. (See Figure 7-22.)
The equation |x – 3| = 7 indicates
that a point on the number line
with a coordinate of x – 3 is 7
units from the origin.
Figure 7-22
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Solve an equation containing a single absolute value expression
Thus, x – 3 can be either 7 or –7.
x–3=7
x = 10
or x – 3 = –7
x = –4
The solutions of |x – 3| = 7 are 10 and –4.
(See Figure 7-23).
Figure 7-23
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Solve an equation containing a single absolute value expression
If either of these numbers is substituted for x in the
equation, it results in a true statement:
|x – 3| = 7
|x – 3| = 7
|10 – 3| ≟ 7
|–4 – 3| ≟ 7
|7| ≟ 7
|–7| ≟ 7
7=7
7=7
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Example 1
Solve: |3x – 2| = 5
Solution:
We can write |3x – 2| = 5 as
3x – 2 = 5
or
3x – 2 = –5
and solve each equation for x:
3x – 2 = 5
or 3x – 2 = –5
3x = 7
3x = –3
x = –1
Verify that both solutions check.
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Solve an equation containing a single absolute value expression
To solve more complicated equations with a term involving
an absolute value, we must isolate the absolute value
before attempting to solve for the variable.
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2.
Solve an equation where two
absolute value expressions are equal
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Solve an equation where two absolute value expressions are equal
The equation |a| = |b| is true when a = b or when a = –b.
For example,
|3| = |3|
|3| = |–3|
3=3
3=3
Thus, we have the following result.
Equations with Two Absolute Values
If a and b represent algebraic expressions, the equation
|a| = |b| is equivalent to the pair of equations
a=b
or
a = –b
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Example 5
Solve: |5x + 3| = |3x + 25|.
Solution:
This equation is true when 5x + 3 = 3x + 25, or when
5x + 3 = –(3x + 25).
We solve each equation for x.
5x + 3 = 3x + 25
or
5x + 3 = –(3x + 25)
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Example 5 – Solution
2x = 22
cont’d
5x + 3 = –3x – 25
x = 11
8x = –28
Verify that both solutions check.
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