Lesson
Notes
Solving Absolute Value Equations
Why?
Then
You solved equations
using properties of
equality. (Lesson 1-3)
Now
Evaluate expressions
involving absolute
values.
Solve absolute value
equations.
NGSSS
Reinforcement of
MA.912.A.3.6 Solve and
graph the solutions of
absolute value equations
and inequalities with one
variable.
New Vocabulary
absolute value
empty set
extraneous solution
FL Math Online
glencoe.com
Sailors sometimes use a laser range
finder to determine distances. Suppose
one such range finder is accurate to
within ±0.5 yard. This means that if
a sailor estimating the distance to
shore reads 323.1 yards on the laser
range finder, the distance to shore
might actually be as close as 322.6 or
as far away as 323.6 yards. These
extremes can be described by the
equation ⎪E - 323.1⎥ = 0.5.
1
FOCUS
Vertical Alignment
Before Lesson 1-4
Solve equations using properties
of equality.
Lesson 1-4
Evaluate expressions involving
absolute values.
Solve absolute value equations.
Absolute Value Expressions The absolute value of a number is its distance from 0 on
the number line. Since distance is nonnegative, the absolute value of a number is always
nonnegative. The symbol |x| is used to represent the absolute value of a number x.
After Lesson 1-4
Solve absolute value inequalities.
Key Concept
Absolute Value
Words
For any real number a, if a is positive or zero, the absolute value of a
is a. If a is negative, the absolute value of a is the opposite of a.
2
Symbols For any real number a, ⎪a⎥ = a if a ≥ 0, and ⎪a⎥ = -a if a < 0.
Model
⎪-4⎥ = 4 and ⎪4⎥ = 4
4 units
-5 -4 -3 -2 -1 0
4 units
1
2
Scaffolding Questions
3
4
Have students read the Why? section of
the lesson.
Ask:
• In ⎪E - 323.1⎥ = 0.5, what does E
represent? the actual distance to
shore
• What is the meaning of the number
0.5 in the equation? the degree of
accuracy of the range finder
• What would be the equation for the
distance to shore estimated at
962.3 yards? ⎪E - 962.3⎥ = 0.5
• How can this be shown on a number
line?
5
When evaluating expressions, absolute value bars act as a grouping symbol. Perform
any operations inside the absolute value bars first.
EXAMPLE 1
Evaluate an Expression with Absolute Value
Evaluate 8.4 - ⎪2n + 5⎥ if n = -7.5.
Replace n with -7.5.
8.4 - ⎪2n + 5⎥ = 8.4 - ⎪2(-7.5) + 5⎥
= 8.4 - ⎪-15 + 5⎥
Multiply 2 and -7.5.
= 8.4 - ⎪-10⎥
Add -15 and 5.
= 8.4 - 10
⎪-10⎥ = 10
= -1.6
Subtract 10 from 8.4.
✓Guided Practice
1
1A. Evaluate ⎪4x + 3⎥ - 3_
if x = -2. 1
2
_1
2
TEACH
E
1
2
1B. Evaluate 1_
- ⎪2y + 1⎥ if y = -_
. 1
3
3
Personal Tutor glencoe.com
Lesson 1-4 Solving Absolute Value Equations
961.8 962.3 962.8
27
Lesson 1-4 Resources
27
Resource
Teacher Edition
027_032_C01_L04_892265.indd
Chapter
Resource
Masters
Transparencies
Other
Approaching-Level
On-Level
11/14/08
3:12:11 PM
Beyond-Level
English Learners
• Differentiated Instruction, p. 29
• Differentiated Instruction, pp. 29, 32
• Differentiated Instruction, p. 32
• Differentiated Instruction, p. 29
• Study Guide and Intervention,
pp. 24–25
• Skills Practice, p. 26
• Practice, p. 27
• Word Problem Practice, p. 28
• Study Guide and Intervention,
pp. 24–25
• Skills Practice, p. 26
• Practice, p. 27
• Word Problem Practice, p. 28
• Enrichment, p. 29
• Spreadsheet Activity, p. 30
• Practice, p. 27
• Word Problem Practice, p. 28
• Enrichment, p. 29
• Study Guide and Intervention,
pp. 24–25
• Skills Practice, p. 26
• Practice, p. 27
• Word Problem Practice, p. 28
• 5-Minute Check Transparency 1-4
• 5-Minute Check Transparency 1-4
• 5-Minute Check Transparency 1-4
• 5-Minute Check Transparency 1-4
• Study Notebook
• Study Notebook
• Study Notebook
• Study Notebook
Lesson 1-4 Solving Absolute Value Equations
0027_0032_C01L04_892270.indd 27
27
12/12/08 1:29:19 PM
Absolute Value Equations Some equations contain absolute value expressions. The
definition of absolute value is used in solving these equations. For any real numbers
a and b, where b ≥ 0, if |a| = b, then a = b or -a = b. This second case is often written
as a = -b.
Absolute Value Expressions
Example 1 shows how to evaluate
expressions that contain absolute
values.
Problem-SolvingTip
✓ Formative Assessment
Write an Equation
Frequently, the best
way to solve a problem
is to use the given
information to write
and solve an equation.
Use the Guided Practice exercises after
each example to determine students’
understanding of concepts.
EXAMPLE 2
Solve an Absolute Value Equation
TENNIS A standard adult tennis racket has a 100-square-inch head, plus or minus
20 square inches. Write and solve an absolute value equation to determine the least
and greatest possible sizes for the head of an adult tennis racket.
Understand We need to determine the greatest and least possible sizes for the head of
a tennis racket given the middle size and the range in sizes.
Plan When writing an absolute value equation, the middle or central value is
always placed inside the absolute value symbols. The range is always
placed on the other side of the equality symbol.
Additional Example
range
central value
1
Evaluate 2.7 + ⎪6 - 2x⎥ if
x = 4. 4.7
⎪x - c⎥ = r
Solve
⎪x - c⎥ = r
Absolute value equation
⎪x - 100⎥ = 20
Additional Examples also in
Interactive Classroom PowerPoint®
Presentations
Case 1
INTERACTIVE
IWB WHITEBOARD
c = 100, and r = 20
a=b
a = -b
Case 2
x - 100 = 20
x - 100 = -20
x - 100 + 100 = 20 + 100
x - 100 + 100 = -20 + 100
x = 120
READY
Check
l the
amples paralle
Additional Ex
.
tly
ac
e text ex
examples in th
ese
th
r
lutions fo
Step-by-step so
in
cluded
examples are in
ssroom.
la
C
Interactive
Originally, players used
leather gloves to hit tennis
balls. Soon after, the glove
was placed at the end of a
stick to extend the reach
of the “hand.”
x = 80
⎪x – 100⎥ = 20
⎪x - 100⎥ = 20
⎪120 – 100⎥ 20
⎪80 - 100⎥ 20
⎪20⎥ 20
⎪-20⎥ 20
20 = 20 ✔
20 = 20 ✔
On a number line, you can see that both solutions are 20 units away
from 100.
20 units
Source: The Cliff Richard Tennis
Foundation
80
90
20 units
100
110
120
The solutions are 120 and 80. The greatest size is 120 square inches and
the least is 80 square inches.
Tips for New Teachers
✓Guided Practice
Reading Students may find it helpful to
read the first absolute value bar as “the
distance of” and the last absolute value
bar as “from zero, without regard to
direction.” So, the expression
⎪6 - 2x⎥ would be read as “the
distance of the value of 6 - 2x from
zero, without regard to direction.”
Solve each equation. Check your solutions.
2A. 9 = ⎪x + 12⎥ {-21, -3}
2B. 8 = ⎪y + 5⎥ {-13, 3}
Personal Tutor glencoe.com
Because the absolute value of a number is always positive or zero, an equation like
|x| = -4 is never true. Thus, it has no solution. The solution set for this type of equation
is the empty set, symbolized by { } or ∅.
28 Chapter 1 Equations and Inequalities
Absolute Value Equations
Examples 2–4 show how to solve
absolute value equations with two
solutions, no solution, and one solution.
027_032_C01_L04_892265.indd
28
11/14/08
3:12:36 PM
Additional Example
2
Solve ⎪y + 3⎥ = 8. Check your
solutions. {–11, 5}
28 Chapter 1 Equations and Inequalities
0027_0032_C01L04_892270.indd 28
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EXAMPLE 3
No Solution
Additional Examples
Solve ⎪3x - 2⎥ + 8 = 1.
⎪3x - 2⎥ + 8 = 1
⎪3x - 2⎥ + 8 - 8 = 1 - 8
⎪3x - 2⎥ = -7
Original equation
3
4
Subtract 8 from each side.
Simplify.
This sentence is never true. The solution set is ∅.
Solve ⎪6 - 4t⎥ + 5 = 0. Solve ⎪8 + y⎥ = 2y - 3. Check
your solutions. {11}
✓Guided Practice
Solve each equation. Check your solutions.
3A. -2⎪3a⎥ = 6 ∅
Focus on Mathematical Content
3B. ⎪4b + 1⎥ + 8 = 0 ∅
Absolute Value The absolute value
of a number is the distance of that
number from 0 on a number line.
Therefore, the statement “the
absolute value of x is always x” is
not true. For example, if x is –3 then
the absolute value of –3 is 3.
Personal Tutor glencoe.com
It is important to check your answers when solving absolute value equations. Even if
the correct procedure for solving the equation is used, the answers may not be actual
solutions to the original equation. Such a number is called an extraneous solution.
StudyTip
EXAMPLE 4
Absolute Value It is
possible for an absolute
value equation to have
only one solution.
Remember to set up
two cases. Then check
your solutions.
One Solution
Solve ⎪x + 10⎥ = 4x - 8. Check your solutions.
a=b
Case 1
a = -b
Case 2
x + 10 = 4x - 8
x + 10 = -(4x - 8)
Watch Out!
x + 10 = -4x + 8
10 = 3x - 8
18 = 3x
Preventing Errors Remind students to
think about the meaning of the
mathematical sentence before they
begin their calculations and again
when they evaluate the reasonableness
of their solution. Explain to students
that they can solve verbal problems
when they ask questions about words
they do not understand, take time to
read, understand, and plan, using a
sketch to help.
5x + 10 = 8
6=x
5x = -2
2
x = -_
5
2
There appear to be two solutions, 6 and -_
.
5
CHECK Substitute each value in the original equation.
⎪x + 10⎥ = 4x - 8
⎪6 + 10⎥ 4(6) - 8
⎪16⎥ 24 - 8
16 = 16 ✔
⎪x + 10⎥ = 4x - 8
⎪-_25 + 10⎥ 4(-_25 ) - 8
⎪9_35 ⎥ -1_35 - 8
3
3
9_
≠ -9_
✘
5
5
3
3
≠ -9_
, the only solution is 6. The solution set is {6}.
Because 9_
5
5
✓Guided Practice
INTERACTIVE WHITEBOARD
On the board, work through
several examples solving
absolute value equations. Save
your work to a file and send it
to your students so they can use
it as an additional reference.
Solve each equation. Check your solutions.
4B. 3⎪2x + 2⎥ - 2x = x + 3 -1
4A. 2⎪x + 1⎥ - x = 3x - 4 3
Personal Tutor glencoe.com
Lesson 1-4 Solving Absolute Value Equations
Differentiated Instruction
027_032_C01_L04_892265.indd
If
Then
29
AL
OL
ELL
29
11/14/08
3:12:45 PM
students found anything from the lesson confusing,
ask them to record two or three of the confusing items separately on an index card. Have
them write an explanation or example for each item in their own words. This will help them
review in the future.
Lesson 1-4 Solving Absolute Value Equations
0027_0032_C01L04_892270.indd 29
29
12/12/08 1:29:32 PM
3
✓ Check Your Understanding
PRACTICE
✓ Formative Assessment
Example 1
p. 27
Use Exercises 1–13 to check for
understanding.
Evaluate each expression if x = -4 and y = -9. 4.
1. ⎪x - 8⎥ 12
4. -2⎪3x + 8⎥ - 4
-12
5. FISH Most freshwater tropical fish thrive if
the water is within 2°F of 78°F.
a. Write an equation to determine the least
and greatest optimal temperatures. ⎪x - 78⎥ = 2
Use the chart at the bottom of this page
to customize assignments for your
students.
80
79
78
77
76
75
74
73
72
b. Solve the equation you wrote in part a.
5b. least: 76°F,
greatest: 80°F
Multiple Representations In
Exercise 44, students use a number line,
information organized in a table, and
algebra to analyze inequalities.
Examples 2–4
pp. 28–29
Watch Out!
69
ºF
Solve each equation. Check your solutions.
6. ⎪x + 8⎥ = 12 {4, -20}
7. ⎪y - 4⎥ = 11 {15, -7}
8. ⎪a - 5⎥ + 4 = 9 {10, 0}
9. ⎪b - 3⎥ + 8 = 3 ∅
{3, 0}
11. -2⎪5y - 1⎥ = -10
12. ⎪a - 4⎥ = 3a - 6 2.5
Error Analysis For Exercise 45,
students should see that Ana and Ling
have differences in their solutions. They
must decide which person checked the
solutions correctly.
71
70
c. If your aquarium’s thermometer is accurate
to within plus or minus 1°F, what should the
temperature of the water be to ensure that it reaches the minimum temperature?
Explain. 77°F; This would ensure a minimum temperature of 76°F.
10. 3⎪2x - 3⎥ - 5 = 4
13. ⎪b + 5⎥ = 2b + 3 2
Example 1
Additional Answers
{_65 , -_45 }
= Step-by-Step Solutions begin on page R20.
Extra Practice begins on page 947.
Practice and Problem Solving
p. 27
Evaluate each expression if a = -3, b = -5, and c = 4.2.
14. ⎪-3c⎥ 12.6
15. ⎪5b⎥ 25
16. ⎪a - b⎥ 2
17. ⎪b - c⎥ 9.2
18. ⎪3b - 4a⎥ 3
19. 2⎪4a - 3c⎥ 49.2
20. -⎪3c - a⎥ -15.6 21. -⎪abc⎥ -63
22. FOOD To make cocoa powder, cocoa beans are roasted. The ideal temperature for
roasting is 300°F, plus or minus 25°. Write and solve an equation describing the
maximum and minimum roasting temperatures for cocoa beans. |x - 300| = 25;
43. ⎪x - 100⎥ = 245; maximum: 345
ft above sea level; minimum: –145
ft below sea level. No, the
maximum is reasonable but the
minimum is not. Florida’s lowest
point should be at sea level where
Florida meets the Atlantic Ocean
and the Gulf of Mexico.
maximum: 325°F; minimum: 275°F
Examples 2–4
pp. 28–29
Solve each equation. Check your solutions.
23. ⎪z - 13⎥ = 21 {34, -8}
24. ⎪w + 9⎥ = 17 {8, -26}
25. 9 = ⎪d + 5⎥ {4, -14}
26. 35 = ⎪x - 6⎥ {-29, 41}
27. 5⎪q + 6⎥ = 20 {-2, -10}
44a. Sample answer:
B
" # $ % '
-3 -2 -1 0 1 2 3
44c. Sample answer: If A is less than B,
then any number added to or
subtracted from A will be less
than the same number added to
or subtracted from B. If B is
greater than A, then any number
added to or subtracted from B is
greater than the same number
added to or subtracted from A.
3. -3⎪xy⎥ -108
2. ⎪7y⎥ 63
28. -3⎪r + 4⎥ = -21 {3, -11}
1
29. 3⎪2a - 4⎥ = 0 2
30. 8⎪5w - 1⎥ = 0
5
9
1
32. 4⎪7y + 2⎥ - 8 = -7 - , 31 2⎪3x - 4⎥ + 8 = 6 ∅
4
28
33. -3⎪3t - 2⎥ - 12 = -6 ∅
34. -5⎪3z + 8⎥ - 5 = -20
- 5 , - 11
3
3
35. MONEY The U.S. Mint produces quarters that weigh about 5.67 grams each. After the
quarters are produced, a machine weighs them. If the quarter weighs 0.02 gram more
or less than the desired weight, the quarter is rejected. Write and solve an equation to
find the heaviest and lightest quarters the machine will approve. ⎪x - 5.67⎥ = 0.02;
_
{ _ _}
{ _ _}
heaviest: 5.69 g; lightest: 5.65 g
Evaluate each expression if q = -8, r = -6, and t = 3.
36. 12 - t⎪3r + 2⎥ -36
37. 2q + ⎪2rt + q⎥ 28
38. -5t - q|8r - t| 393
30 Chapter 1 Equations and Inequalities
Differentiated Homework Options
027_032_C01_L04_892265.indd
Level
30
Assignment
11/14/08
3:12:50 PM
Two-Day Option
AL
Basic
14–34, 45, 47–74
15–33 odd, 52–55
14–34 even, 45, 47–51,
56–74
OL
Core
15–43 odd, 44, 45, 47–74
14–34, 52–55
35–45, 47–51, 56–74
BL
Advanced
35–68, (optional: 69–74)
30 Chapter 1 Equations and Inequalities
0027_0032_C01L04_892270.indd 30
12/12/08 1:29:40 PM
Study Guide and Intervention
pp. 24–25 AL OL ELL
Solve each equation. Check your solutions.
4
1
39. 8x = 2⎪6x - 2⎥ 1,
40. -6y + 4 = ⎪4y + 12⎥ 5
5
8
23
41. 8z + 20 = -⎪2z + 4⎥ 42. -3y - 2 = ⎪6y + 25⎥ -3, 3
3
_
{
{ _}
_
43
• Words
For any real number a, if a is positive or zero, the absolute value of a is a.
If a is negative, the absolute value of a is the opposite of a.
• Symbols
For any real number a, ⎪a⎥ = a, if a ≥ 0, and ⎪a⎥ = -a, if a < 0.
Absolute Value
Example 1
if x = 6.
Example 2
Evaluate 2x - 3y if
x = -4 and y = 3.
Evaluate -4 - -2x ⎪-4⎥ - ⎪-2x⎥
MULTIPLE REPRESENTATIONS Draw a number line. a. See margin.
⎪2x - 3y⎥ = ⎪2(-4) - 3(3)⎥
= ⎪-4⎥ - ⎪-2 6⎥
= ⎪-4⎥ - ⎪-12⎥
= 4 - 12
= -8
= ⎪-8 - 9⎥
= ⎪-17⎥
= 17
Exercises
1
Evaluate each expression if w = -4, x = 2, y = −
, and z = -6.
2
< B
A ____
> A+C
B + C ____
> A+D
B + D ____
> A+F
B + F ____
> A
B ____
> A
B ____
1
5. ⎪x⎥ - ⎪y⎥ - ⎪z⎥ -4−
6. ⎪7 - x⎥ + ⎪3x⎥ 11
7. ⎪w - 4x⎥ 12
8. ⎪wz⎥ - ⎪xy⎥ 23
> A-C
B - C ____
> A-D
B - D ____
> A-F
B - F ____
12. 10 - ⎪xw⎥ 2
13. ⎪6y + z⎥ + ⎪yz⎥ 6
1
14. 3⎪wx⎥ + −
⎪4x + 8y⎥ 27
4
15. 7⎪yz⎥ - 30
16. 14 - 2⎪w - xy⎥ 4
17. ⎪2x - y⎥ + 5y 6
18. ⎪xyz⎥ + ⎪wxz⎥ 54
19. z⎪z⎥ + x⎪x⎥ -32
20. 12 - ⎪10x - 10y⎥ -3
1⎪
5z + 8w⎥ 31
21. −
2
22. ⎪yz - 4w⎥ -w 17
3⎪ ⎥
1
23. −
wz + −
⎪8y⎥ 20
4
2
24. xz - ⎪xz⎥
24
Practice
p. 27
005_042_A2CRMC01_890526.indd 24
If B > A, then B + x > A + x. If B > A, then B - x > A - x.
AL
1-4
Use Higher-Order Thinking Skills
4/11/08 12:50:20 AM
ELL
Practice
Solving Absolute Value Equations
1. |6a|
45. ERROR ANALYSIS Ana and Ling are solving ⎪3x + 14⎥ = -6x. Is either of them correct?
Explain your reasoning.
Ana
Ling
| 3x + 14| = –6x
3x + 14 = –6x or 3x + 14 = 6x
9x = –14
14 = 3x
14
14
✔
x=_ ✔
x = –_
9
|3x + 14| = –6x
3x + 14 = –6x or 3x + 14 = 6x
9x = –14
14 = 3x
14
14
✗
x=_
x = –_
9
3 ✔
6
2. |2b + 4|
12
3. - |10d + a| -15
4. |17c| + |3b - 5|
5. -6 |10a - 12| -132
6. |2b - 1| - |-8b + 5|
7. |5a - 7| + |3c - 4| 23
8. |1 - 7c| - |a|
9. -3|0.5c + 2| - |-0.5b| -17.5
11. |a - b| + |b - a| 14
114
-52
33
10. |4d| + |5 - 2a|
12.6
12. |2 - 2d| - 3|b|
-19.2
Solve each equation. Check your solutions.
13. |n - 4| = 13
46. CHALLENGE Solve ⎪2x - 1⎥ + 3 = ⎪5 - x⎥. List all cases and resulting equations.
(Hint: There are four possible cases to examine as potential solutions.) See Chapter 1
Answer Appendix.
REASONING If a, x, and y are real numbers, determine whether each statement is
sometimes, always, or never true. Explain your reasoning.
{-9, 17}
14. |x - 13| = 2
{11, 15}
15. |2y - 3| = 29
{-13, 16}
16. 7|x + 3| = 42
17. |3u - 6| = 42
{-12, 16}
18. |5x - 4| = -6 ∅
{-9, 3}
{1.75, 3.25}
19. -3 |4x - 9| = 24
∅
20. -6|5 - 2y| = -9
21. |8 + p| = 2p - 3
{11}
22. |4w - 1| = 5w + 37 {-38}
23. 4 |2y - 7| + 5 = 9
25. 2 |4 - s| = -3s
{3, 4}
24. -2|7 - 3y| - 6 = -14
{-8}
27. 5 |2r + 3| - 5 = 0
{-2, -1}
{1, 3 −23 }
{-3, 1}
26. 5 - 3|2 + 2w| = -7
28. 3 - 5|2d - 3| = 4 ∅
29. WEATHER A thermometer comes with a guarantee that the stated temperature
differs from the actual temperature by no more than 1.5 degrees Fahrenheit. Write
and solve an equation to find the minimum and maximum actual temperatures when
the thermometer states that the temperature is 87.4 degrees Fahrenheit.
x - 87.4 ≤ 1.5; or 85.9 ≤ x ≤ 88.9
30. OPINION POLLS Public opinion polls reported in newspapers are usually given with a
margin of error. For example, a poll with a margin of error of ±5% is considered accurate
to within plus or minus 5% of the actual value. A poll with a stated margin of error of
63% predicts that candidate Tonwe will receive 51% of an upcoming vote. Write and solve
an equation describing the minimum and maximum percent of the vote that candidate
Tonwe is expected to receive.
47. If ⎪a⎥ > 7, then ⎪a + 3⎥ > 10.
x - 51 ≤ 3 or 48 ≤ x ≤ 54
48. If ⎪x⎥ < 3, then ⎪x⎥ + 3 > 0.
27
Chapter 1
49. If y is between 1 and 5, then ⎪y - 3⎥ ≤ 2.
Word Problem Practice
p. 28 AL OL BL
Glencoe Algebra 2
005_042_A2CRMC01_890526.indd 27
50. OPEN ENDED Write an absolute value equation of the form ⎪ax + b⎥ = cx + d that has
no solution. Assume that a, b, c, and d ≠ 0.
1-4
4/11/08 12:50:33 AM
ELL
Word Problem Practice
Solving Absolute Value Equations
51. WRITING IN MATH Explain step by step how you solve an absolute value equation of
the form a⎪x - b⎥ + c = d for x. See Chapter 1 Answer Appendix.
1. LOCATIONS Identical vacation
cottages, equally spaced along a street,
are numbered consecutively beginning
with 10. Maria lives in cottage #17.
Joshua lives 4 cottages away from
Maria. If n represents Joshua’s cottage
number, then |n - 17| = 4. What are
the possible numbers of Joshua’s
cottage?
31
4. TOLERANCE Martin makes exercise
weights. For his 10-pound dumbbells, he
guarantees that the actual weight of his
dumbbells is within 0.1 pound of
10 pounds. Write and solve an equation
that describes the minimum and
maximum weight of his 10-pound
dumbbells.
⎪w - 10⎥ = 0.1
minimum weight: 9.9 pounds
maximum weight: 10.1 pounds
Maria’s
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Glencoe Algebra 2
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-9
Evaluate each expression if a = -1, b = -8, c = 5, and d = -1.4.
Lesson 1-4 Solving Absolute Value Equations
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9. ⎪z⎥ - 3 ⎪5yz⎥ -39
11. ⎪z⎥ - 4 ⎪2z + y⎥ -40
Chapter 1
d. ALGEBRAIC Describe the patterns algebraically, using the variable x to replace
C, D, and F. If A < B, then A + x < B + x. If A < B, then A - x < B - x.
3
2
10. 5 ⎪w⎥ + 2⎪z - 2y⎥ 34
c. VERBAL Describe the patterns in the table. See margin.
H.O.T. Problems
3. 5 + ⎪w + z⎥ 15
4. ⎪x + 5⎥ - ⎪2w⎥ -1
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
45. Ling; Ana included
an extraneous solution.
She would have caught
this error if she had
checked to see if her
answers were correct
by substituting the
values into the
original equation.
47. Sometimes; this is
only true for certain
values of a. For
example, it is true for
a = 8; if 8 > 7, then
11 > 10. However it is
not true for a = -8; if
8 > 7, then 5 ≯ 10.
48. Always; if ⎪x⎥ < 3,
then x is between -3 or
3. Adding 3 to the
absolute value of any of
the numbers in this set
will produce a positive
number.
49. Always; starting
with numbers between 1
and 5 and subtracting 3
will produce numbers
between -2 and 2.
These all have an
absolute value less than
or equal to 2.
50. Sample answer:
⎪2x + 1⎥ = x - 3, or
⎪3x + 10⎥ = x - 5, or
< B
A ____
< B-C
A - C ____
< B-D
A - D ____
< B-F
A - F ____
2. ⎪6 + z⎥ - ⎪-7⎥ -7
Lesson 1-4
< B+C
A + C ____
< B+D
A + D ____
< B+F
A + F ____
1. ⎪2x - 8⎥ 4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. TABULAR Fill in each blank in the table with either > or < using the points from
the number line.
Source: Environmental
Protection Agency
_
Solving Absolute Value Equations
The absolute value of a number is its distance from 0
on a number line. The symbol ⎪x⎥ is used to represent the absolute value of a number x.
C a. GEOMETRIC Label any 5 integers on the number line points A, B, C, D, and F.
During the past century,
sea levels along the
Mid-Atlantic and Gulf
coasts have risen about
6 inches more than the
global average.
Study Guide and Intervention
Absolute Value Expressions
SEA LEVEL Florida is on average 100 feet above sea level. This level varies by as much
as 245 feet depending on precipitation and your location. Write and solve an equation
describing the maximum and minimum sea levels for Florida. Is this solution
reasonable? Explain. See margin.
44.
⎪x - 1⎥ = 1 x - 4
2
1-4
_}
12
14
16
18
20
22
24
5. WALKING Jim is walking along a
straight line. An observer watches him.
If Jim walks forward, the observer
records the distance as a positive
number, but if he walks backward, the
observer records the distance as a
negative number. The observer has
recorded that Jim has walked a, then b,
then c feet.
13 or 21
Enrichment
p. 29 OL
1-4
11/14/08
BL
Enrichment
Considering All Cases in Absolute Value Equations
You have learned that absolute value equations with one set of absolute value
symbols have two cases that must be considered. For example, | x + 3 | = 5 must
be broken into x + 3 = 5 or -(x + 3) = 5. For an equation with two sets of
absolute value symbols, four cases must be considered.
Consider the problem | x + 2 | + 3 = | x + 6 |. First we must write the equations
for the case where x + 6 ≥ 0 and where x + 6 < 0. Here are the equations for
these two cases:
|x + 2| + 3 = x + 6
| x + 2 | + 3 = -(x + 6)
Each of these equations also has two cases. By writing the equations for both
cases of each equation above, you end up with the following four equations:
x+2+3=x+6
-(x + 2) + 3 = x + 6
x + 2 + 3 = -(x + 6)
-x - 2 + 3 = -(x + 6)
3:12:57 PM
2. HEIGHT Sarah and Jessica are sisters.
Sarah’s height is s inches and Jessica’s
height is j inches. Their father wants
to know how many inches separate
the two. Write an equation for this
difference in such a way that the result
will always be positive no matter which
sister is taller.
a. Write a formula for the total distance
that Jim walked.
T = ⎪a⎥ + ⎪b⎥ + ⎪c⎥
d = ⎪s - j ⎥ or d = ⎪j - s⎥
b. The equation you wrote in part a
should not be T = |a + b + c|.
What does |a + b + c| represent?
3. AGES In 2005, 24.8% of all Americans
were under 18 years old. Rhonda
conducts a survey of the ages of
students in eleventh grade at her school.
On November 1, she finds the average
age is 200 months. She also finds that
two-thirds of the students are within
6 months of the average age. Write and
solve an equation to determine the age
limits for this group of students. How
many months will it be till the first of
these students turn 18?
The distance Jim ends up from
where he started.
c. When would the formula you wrote
in part a give the same value as the
formula shown in part b?
They will be equal only if Jim
walks in the same direction
each time giving a, b, and c
all the same sign.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
and
A Study Guide
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10
⎪a - 200⎥ = 3
a = 197 or 203
Solve each of these equations and check your solutions in the original equation,
c.
5
| x + 2 | + 3 = | x + 6 |. The only solution to this equation is - −
.
2
E
Chapter 1
28
Glencoe Algebra 2
i
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Lesson 1-4 Solving Absolute Value Equations
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4
NGSSS PRACTICE
ASSESS
912.A.3.14, 912.P.1.2, 912.A.3.1, 912.G.1.1
52. If 4x - y = 3 and 2x + 3y = 19, what is the value
of y? D
Crystal Ball Have students write how
they think what they have learned in
Lessons 1-3 and 1-4 will connect with
Lesson 1-5, Solving Inequalities.
A.
B.
C.
D.
Tips for New Teachers
53.
Looking Ahead Lesson 1-5 presents
solving inequalities using steps similar
to those for solving equations.
Exercises 69–74 should be used to
determine students’ familiarity with
solving equations.
54. Which equation is equivalent to
4(9 - 3x) = 7 - 2(6 - 5x)? G
2
3
4
5
F.
G.
H.
I.
GRIDDED RESPONSE Two male and 2 female
students from each of the 9th, 10th, 11th, and
12th grades comprise the Student Council. If a
Student Council representative is chosen at
random to attend a board meeting, what is the
probability that the student will be either an
11th grader or male? 5/8
8x = 41
22x = 41
8x = 24
22x = 24
55. SAT/ACT A square with side length 4 units has
one vertex at the point (1, 2). Which one of the
following points cannot be diagonally opposite
that vertex? C
A. (-3, -2)
B. (-3, 6)
C. (5, -3)
D. (5, 6)
Spiral Review
Solve each equation. Check your solution. (Lesson 1-3)
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New teachers
ay
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eciate the Tips
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New Teachers
3
2
58. _y – 7 = _
y + 3 50
57. 5p - 10 = 4(7 + 6p) -2
56. 4x + 6 = 30 6
5
5
59. MONEY Nhu is saving to buy a car. In the first 6 months, his savings were $80 less
3
1
than _
the price of the car. In the second six months, Nhu saved $50 more than _
the
5
4
price of the car. He still needs $370. (Lesson 1-3)
a. What is the price of the car? $6800
b. What is the average amount of money Nhu saved each month? $535.83
c. If Nhu continues to save the average amount each month, in how many months
will he be able to afford the car? 1 mo
Name the property illustrated by each equation. (Lesson 1-2)
60. (1 + 8) + 11 = 11 + (1 + 8) Comm. (+)
61. z(9 - 4) = z · 9 - z · 4 Distributive
Simplify each expression. (Lesson 1-2)
62. 7a + 3b - 4a - 5b 3a - 2b
63. 3x + 5y + 7x - 3y 10x + 2y
64. 3(15x - 9y) + 5(4y - x) 40x - 7y
65. 2(10m - 7a) + 3(8a - 3m) 11m + 10a
66. 8(r + 7t) - 4(13t + 5r) -12r + 4t
67. 4(14c - 10d) - 6(d + 4c) 32c - 46d
68. GEOMETRY The formula for the surface area of a rectangular prism is
SA = 2w + 2h + 2wh, where represents the length, w represents the
width, and h represents the height. Find the surface area of the rectangular
prism at the right. (Lesson 1-1) 358 in2
7 in.
5 in.
12 in.
Skills Review
Solve each equation. (Lesson 1-3)
69. 15x + 5 = 35 2
72. 3(w - 1) = 2w - 6 -3
70. 2.4y + 4.6 = 20 ≈6.417
4
1
73. _(2b - 4) = 2 + 8b 2
7
71. 8a + 9 = 6a - 7 -8
_
1
74. _(6p - 24) = 18 + 3p -26
3
32 Chapter 1 Equations and Inequalities
Differentiated Instruction
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3:13:08 PM
Extension For equations with one set of absolute value symbols, two cases must be considered.
For an equation with two sets of absolute value symbols, four cases must be considered. How many
cases must be considered for an equation containing three sets of absolute value symbols? 8
32 Chapter 1 Equations and Inequalities
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