Download of a variable expression. You can use the definition

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of a variable expression.
You can use the definition of absolute value to solve absolute value
equations algebraically.
Since
then the equation |ax + b| = c is equivalent to í(ax + b) = c or (ax + b) = c.
Since í(ax + b) = c is equivalent to ax + b = íc, the absolute value
equation |ax + b| = c is equivalent to ax + b = íc or ax + b = c.
Example A
Math Tip
Recall that the geometric interpretation of |x| is the distance from the
number x to 0 on a number line.
If |x| = 5, then x = í5 or x = 5 because those two values are both 5
units away from 0 on a number line.
Solve 2|x í 1| í 5 = 1. Graph the solutions on a number line.
Step 1:
Isolate the absolute value expression. Add 5 to both sides and then divide
by 2.
Step 2:
Write and solve two equations using the definition of absolute value.
Solution: There are two solutions: x = 4 and x = í2
Check to see if both solutions satisfy the original equation. Substitute 4
and í2 for x in the original equation.
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To graph the solutions, plot points at 4 and í2 on a number line.
Try These A
Solve each absolute value equation. Graph the solutions on a number
line.
a. |x í 2| = 3
x = 5, x = í1
b. |x + 1| í 4 = í2
x = 1, x = í3
c. |x í 3| + 4 = 4
x=3
d. |x + 2| + 3 = 1
no solution
p. 12
1. Reason abstractly. How many solutions are possible for an
absolute value equation having the form |ax + b| = c, where a, b,
and c are real numbers?
There are possibly two, one, or zero solutions to an absolute value
equation having this form.
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Example B
Math Tip
You know that the distance from t to 82°F on a thermometer is
4.5°F. This distance can be modeled with the absolute value
expression |t í 82|.
The temperature of the wave pool at Sapphire Island can vary up to
4.5°F from the target temperature of 82°F. Write and solve an
absolute value equation to find the temperature extremes of the wave
pool. (The temperature extremes are the least and greatest possible
temperatures.)
Step 1:
Write an absolute value equation to represent the situation.
Let t represent the temperature extremes of the wave pool in degrees
Fahrenheit.
Step 2:
Use the definition of absolute value to solve for t.
Solution: The greatest possible temperature of the wave pool is
86.5°F, and the least possible temperature is 77.5°F. Both of these
temperatures are 4.5°F from the target temperature of 82°F.
Try These B
The pH of water is a measure of its acidity. The pH of the water on
the Seal Slide can vary up to 0.3 from the target pH of 7.5. Use this
information for parts a–c.
a. Write an absolute value equation that can be used to find the
extreme pH values of the water on the Seal Slide. Be sure to
explain what the variable represents.
|p í 7.5| = 0.3; p represents the extreme pH values of the water.
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b. Solve your equation, and interpret the solutions.
p = 7.8, p = 7.2; The greatest allowed pH is 7.8, and the least
allowed pH is 7.2.
c. Reason quantitatively. Justify the reasonableness of your
answer to part b.
Sample answer: 7.8 is 0.3 more than 7.5; 7.2 is 0.3 less than 7.5.
p. 13
Math Terms
An absolute value inequality is an inequality involving the
absolute value of a variable expression.
Solving absolute value inequalities algebraically is similar to
solving absolute value equations. By the definition of absolute
value, |ax + b| > c, where c > 0, is equivalent to í(ax + b) > c or ax
+ b > c. Multiplying the first inequality by í1, and then using a
similar method for |ax + b| < c, gives these statements:
|ax + b| > c, c > 0, is equivalent to ax + b < íc or ax + b > c.
|ax + b| < c, c > 0, is equivalent to ax + b < c or ax + b > íc,
which can also be written as íc < ax + b < c.
Example C
Solve each inequality. Graph the solutions on a number line.
Math Tip
These properties of real numbers can help you solve
inequalities. The properties also apply to inequalities that
include <, •, or ”.
Addition Property of Inequality
If a > b, then a + c > b + c.
Subtraction Property of Inequality
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If a > b, then a í c > b í c.
Multiplication Property of Inequality
If a > b and c > 0, then ca > cb.
If a > b and c < 0, then ca < cb.
Division Property of Inequality
If a > b and c > 0, then
.
If a > b and c < 0, then
.
a. |2x + 3| + 1 > 6
Step 1:
Isolate the absolute value expression.
Step 2:
Write two inequalities.
Step 3:
Solve each inequality.
Solution:
b. |3x í 1| + 5 < 7
Step 1:
Isolate the absolute value expression.
Step 2:
Write the compound inequality.
Step 3:
Solve the inequality.
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Solution:
Try These C
Solve and graph each absolute value inequality.
a.
x > 5 or x < í1
b.
í4 ” x ” 0
c.
x • 1 or
d.
í6 < x < í1
p. 14
2. Make sense of problems. Why is the condition c > 0
necessary for |ax + b| < c to have a solution?
If c = 0 or if c < 0 (c is negative), the inequality would be
impossible or trivial because absolute value is the distance
from zero on a number line.
Check Your Understanding
Academic Vocabulary
When you compare and contrast two topics, you describe
ways in which they are alike and ways in which they are
different.
3. Compare and contrast a linear equation having the form
ax + b = c with an absolute value equation having the
form |ax + b| = c.
Sample answer: A linear equation of the form ax + b = c has 1
solution, but an absolute value equation of the form |ax + b| =
c may have 0, 1, or 2 solutions depending on the value of c.
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4. Critique the reasoning of others. Paige
incorrectly solved an absolute value equation as
shown below.
a. What mistake did Paige make?
She did not isolate the absolute value expression
first.
b. How could Paige have determined that her
solutions are incorrect?
Sample answer: She could have substituted the
values for x into the original equation to check
whether they satisfy the equation.
c. Solve the equation correctly. Explain your steps.
No solution; Sample explanation: First, isolate the
absolute value expression: |x + 5| = –4. The equation
|x + 5| = –4 has no solutions because the absolute
value of an expression cannot be negative.
5. Explain how to write the inequality |5x í 6| • 9 without
using an absolute value expression.
Sample answer: The inequality has the form |ax – b| • c,
where c • 0, so it can be written as ax – b • c or ax – b ”
–c. Thus, |5x – 6| • 9 is equivalent to 5x – 6 • 9 or 5x – 6 ”
–9.
Lesson 1-3 Practice
Solve each absolute value equation.
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6. |x í 6| = 5
x = 1, x = 11
7. |3x í 7| = 12
8. |2x + 9| í 10 = 5
x = 3, x = –12
9. |5x í 3| + 12 = 4
no solution
10. Model with mathematics. The flow rate on the
Otter River Run can vary up to 90 gallons per
minute from the target flow rate of 640 gallons per
minute. Write and solve an absolute value equation
to find the extreme values of the flow rate on the
Otter River Run.
|f – 640| = 90, where f represents the extreme flow rates
in gallons per minute; f = 730, f = 550; The greatest flow
rate is 730 gallons per minute and the least flow rate is
550 gallons per minute.
Solve each absolute value inequality. Graph the solutions
on a number line.
11. |x í 7| > 1
x < 6 or x > 8
12. |2x í 5| ” 9
–2 ” x ” 7
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13. |3x í 10| í 5 • í1
x ” 2 or
14. |4x + 3| í 9 < 5
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