3-6
3-6
Equations and Problem Solving
1. Plan
Objectives
1
2
To define a variable in terms
of another variable
To model distance-rate-time
problems
Examples
1
2
3
4
5
Defining One Variable in
Terms of Another
Consecutive Integer Problem
Same-Direction Travel
Round-Trip Travel
Opposite-Direction Travel
Math Background
What You’ll Learn
Check Skills You’ll Need
• To define a variable
Write a variable expression for each situation.
in terms of another
variable
Lesson 1-1
2. twice the length O 2/
1. value in cents of q quarters 25q
3. number of miles traveled at 34 mi/h in h hours 34h
4. weight of 5 crates if each crate weighs x kilograms 5x
• To model distance-rate-time
problems
. . . And Why
5. cost of n items at $3.99 per item 3.99n
To solve real-world problems
involving distance, rate, and
time, as in Examples 3–5
2-1
GO for Help
New Vocabulary • consecutive integers • uniform motion
Defining
Part 1 Variables
Using Equations to Solve Problems
Some problems contain two or more unknown quantities. To solve such problems,
first decide which unknown quantity the variable will represent. Then express the
other unknown quantity or quantities in terms of that variable.
Time-motion relationships occur
in the real world in situations
ranging from the atomic to the
astronomic. Students will use the
concepts of the lesson in much
future work in mathematics.
1
EXAMPLE
Defining One Variable in Terms of Another
Geometry The length of a rectangle is 6 in. more than its width. The perimeter of
the rectangle is 24 in. What is the length of the rectangle?
More Math Background: p. 116D
Relate The length is 6 in. more than the width.
Lesson Planning and
Resources
Problem Solving Hint
For Example 1,
drawing a diagram
will help you
understand the
problem.
See p. 116E for a list of the
resources that support this lesson.
Then w + 6 = the length.
Write
w⫹6
PowerPoint
Bell Ringer Practice
Define Let w = the width.
w
P = 2O + 2w
Use the perimeter formula.
24 = 2(w + 6) + 2w
Substitute 24 for P and w ± 6 for O.
24 = 2w + 12 + 2w
Use the Distributive Property.
24 = 4w + 12
Combine like terms.
24 - 12 = 4w + 12 - 12
Check Skills You’ll Need
For intervention, direct students to:
Using Variables
Simplify.
12 = 4w
4
4
Divide each side by 4.
Simplify.
The width of the rectangle is 3 in. The length of the rectangle is 6 in. more than the
width. So the length of the rectangle is 9 in.
Quick Check
158
1 The width of a rectangle is 2 cm less than its length. The perimeter of the rectangle
is 16 cm. What is the length of the rectangle? 5 cm
Chapter 3 Solving Equations
Special Needs
Below Level
L1
Have students always draw a diagram or make a table
to model the problem. Help them identify what it is
they want to know, assign a variable to that
unknown, and set up the equation before beginning
calculations.
158
Subtract 12 from each side.
12 = 4w
3=w
Lesson 1-1: Example 1
Extra Skills and Word
Problem Practice, Ch. 1
The length is described in terms of the width.
So define a variable for the width first.
learning style: visual
L2
Emphasize how helpful it is for students to organize
the information from a distance-rate-time problem
into a table before attempting to solve the problem.
learning style: verbal
Consecutive integers differ by 1. The integers 50 and 51 are consecutive integers,
and so are -10, -9, and -8. For consecutive integer problems, it may help to define a
variable before describing the problem in words. Let a variable represent one of the
unknown integers. Then define the other unknown integers in terms of the first one.
2
Consecutive Integer Problem
EXAMPLE
Relate
Write
n
plus
+
n+ 1
2
plus
third
integer
is
147
+
n+ 2
=
147
n + n + 1 + n + 2 = 147
3n + 3 = 147
Combine like terms.
3n + 3 - 3 = 147 - 3
Subtract 3 from each side.
3n = 144
Simplify.
3n = 144
3
3
Divide each side by 3.
n = 48
EXAMPLE
Math Tip
Defining the variable is a key step
in interpreting the solution of the
equation.
Define Let n = the first integer.
Then n + 1 = the second integer,
and n + 2 = the third integer.
second
integer
Guided Instruction
1
The sum of three consecutive integers is 147. Find the integers.
first
integer
2. Teach
Simplify.
If n = 48, then n + 1 = 49, and n + 2 = 50. The three integers are 48, 49, and 50.
EXAMPLE
Visual Learners
Make sure students understand
consecutive integers. Have them
look at 4 and 5 on a number line.
Ask students to give their own
definition of consecutive integers
using the number line. On a
number line, consecutive
integers do not have any other
integers between them. Ask
students to name three
consecutive integers. Answers
may vary. Sample: 7, 8, 9
Check Is the solution correct? Yes; 48 + 49 + 50 = 147.
Quick Check
2 The sum of three consecutive integers is 48.
a. Define a variable for one of the integers. Let x ≠ the first integer.
b. Write expressions for the other two integers. x ± 1 is the second integer
and x ± 2 is
c. Write and solve an equation to find the three integers.
3x + 3 = 48; 15, 16, 17
the third integer.
2
1 2 Distance-Rate-Time Problems
Part
An object that moves at a constant rate is said to be in uniform motion.
The formula d = rt gives the relationship between distance d, rate r, and time t.
Uniform motion problems may involve objects going the same direction, opposite
directions, or round trips.
nline
In the diagram below, the two vehicles are traveling the same direction at different
rates. The distances the vehicles travel are the same.
40 mi/h • 5 h
Visit: PHSchool.com
Web Code: ate-0775
200 mi
50 mi/h • 4 h
Since the distances are equal, the products of rate and time for the two cars are
equal. For the vehicles shown, 40 ? 5 = 50 ? 4.
Lesson 3-6 Equations and Problem Solving
Advanced Learners
159
English Language Learners ELL
L4
Remind students that the units of distance used to
solve a problem must be consistent.
learning style: verbal
Make sure students start by identifying what they
want to find. Students write and complete the
sentence Let x = <name of what they want to find>.
Have students explain the problem in their own words
(using diagrams or tables).
learning style: verbal
159
PowerPoint
A table can also help you understand relationships in distance-rate-time
problems.
Additional Examples
1 The width of a rectangle is
3 in. less than its length. The
perimeter of the rectangle is
26 in. What is the width of the
rectangle? 5 in.
3
Engineering A train leaves a train station at 1 P.M. It travels at an average rate
of 72 mi/h. A high-speed train leaves the same station an hour later. It travels at
an average rate of 90 mi/h. The second train follows the same route as the first
train on a track parallel to the first. In how many hours will the second train
catch up with the first train?
2 The sum of three consecutive
integers is 72. Find the integers.
23, 24, 25
3
EXAMPLE
Define Let t = the time the first train travels.
Then t - 1 = the time the second train travels.
Tactile Learners
Since each distance is represented
by a different expression, some
students may not understand how
the distances can be the same.
Have two students demonstrate.
Both students stand at the same
starting place. One student starts
walking very slowly. The second
student waits a few moments.
Then the second student starts
walking quickly to catch up with
the first student. Show how they
have traveled the same distance
but at different speeds or rates.
Same-Direction Travel
EXAMPLE
Relate
Write
Real-World
Connection
High-speed trains that go
from Boston to New York in
less than 4 hours can reach a
speed of 150 mi/h.
Train
Rate
Time
Distance Traveled
1
72
t
72t
2
90
t⫺1
90(t ⫺ 1)
72t = 90(t - 1) The distances traveled by the trains are equal.
Method 1 Solve by using a table.
Use a table to evaluate each side of the equation. Look for matching values.
Time
72t
90(t ⴚ 1)
1
72(1) = 72
90(1 – 1) = 0
2
72(2) = 144
90(2 – 1) = 90
3
72(3) = 216
90(3 – 1) = 180
4
72(4) = 288
90(4 – 1) = 270
5
72(5) = 360
90(5 – 1) = 360 ✓
PowerPoint
Additional Example
3 An airplane left an airport
flying at 180 mi/h. A jet that flies
at 330 mi/h left 1 hour later. The
jet follows the same route as the
airplane at a different altitude.
How many hours will it take the
jet to catch up with the airplane?
115 h
When the first train travels 5 hours, the second train travels 4 hours (t – 1). The
second train will catch up with the first train in 4 hours.
Method 2 Solve the equation.
72t = 90t - 90
72t - 72t = 90t - 90 - 72t
Use the Distributive Property to simplify 90 (t – 1).
Subtract 72t from each side.
0 = 18t - 90
Combine like terms.
0 + 90 = 18t - 90 + 90
Add 90 to each side.
90 = 18t
Simplify.
90 = 18t
18
18
Divide each side by 18.
t=5
t-1=4
Simplify.
Find the time the second train travels.
The second train will catch up with the first train in 4 h.
Quick Check
3 A group of campers and one group leader left a campsite in a canoe. They traveled
at an average rate of 10 km/h. Two hours later, the other group leader left the
campsite in a motorboat. He traveled at an average rate of 22 km/h.
a. How long after the canoe left the campsite did the motorboat catch up with it?
b. How long did the motorboat travel? 12 h
32
3h
3
160
160
Chapter 3 Solving Equations
PowerPoint
For uniform motion problems that involve a round trip, it is important to
remember that the distance going is equal to the distance returning.
Additional Examples
4 Suppose you hike up a hill at
4 km/h. You hike back down at
6 km/h. Your hiking trip took
3 hours. How much time did it
take you to hike up the hill? 1.8 h
20 mi/h • 3 h
60 mi
30 mi/h • 2 h
60 mi
Since the distances are equal, the products of rate and time for traveling in both
directions are equal. That is, 20 ? 3 = 30 ? 2.
4
EXAMPLE
Round-Trip Travel
Noya drives into the city to buy a software program at a computer store. Because
of traffic conditions, she averages only 15 mi/h. On her drive home she averages
35 mi/h. If the total travel time is 2 hours, how long does it take her to drive to the
computer store?
Define Let t = time of Noya’s drive to the computer store.
Problem Solving Hint
The total travel time is
for a round trip. If it
takes t out of a 2-hour
round trip to get to
the store, then 2 – t is
the time it will take
for the drive home.
2 - t = the time of Noya’s drive home.
Relate
Write
Part of Noya’s Travel
Rate
Time
Distance
To the computer store
15
t
15t
Return home
35
2⫺t
35(2 ⫺ t)
Noya drives
15t miles to the
computer store and
35(2 ⴚ t) miles back.
15t = 35(2 - t)
The distances traveled to and from the store are equal.
15t = 70 - 35t
Use the Distributive Property.
15t + 35t = 70 - 35t + 35t
Add 35t to each side.
50t = 70
Combine like terms.
50t = 70
50
50
Divide each side by 50.
t = 1.4
Simplify.
It took Noya 1.4 h to drive to the computer store.
Quick Check
4 On his way to work from home, your uncle averaged only 20 miles per hour. On his
drive home, he averaged 40 miles per hour. If the total travel time was 1 12 hours,
how long did it take him to drive to work? 1h
For uniform motion problems involving two objects moving in opposite directions,
you can write equations using the fact that the sum of their distances is the total
distance.
5 Two jets leave Dallas at the
same time and fly in opposite
directions. One is flying west
50 mi/h faster than the other.
After 2 hours, the jets are
2500 miles apart. Find the speed
of each jet.
jet flying east: 600 mi/h;
jet flying west: 650 mi/h
4
EXAMPLE
Error Prevention
Students may write t - 2 instead
of 2 - t. Ask: Which is greater,
the total travel time or the time
to drive to the computer store?
total travel time Which makes
more sense, t - 2 or 2 - t? 2 – t
5
EXAMPLE
Tactile Learner
Make a number line on the floor
with masking tape. Have two
students stand beside each other
at 0. Instruct the students to
walk in opposite directions for
2 seconds. Have one student walk
very slowly and one walk briskly.
Lead students to note that the
two distances are not equal, but
combine to make a total distance
walked.
Resources
• Daily Notetaking Guide 3-6 L3
• Daily Notetaking Guide 3-6—
L1
Adapted Instruction
Closure
9 mi
Ask: What are two ways to help
set up distance problems like the
ones in this lesson? Answers may
vary. Sample: draw a diagram
and use a table
3 mi
6 mi
Lesson 3-6 Equations and Problem Solving
161
161
3. Practice
5
Jane and Peter leave their home traveling in opposite directions on a straight road.
Peter drives 15 mi/h faster than Jane. After 3 hours, they are 225 miles apart. Find
Peter’s rate and Jane’s rate.
Assignment Guide
1 A B 1-8, 15-16, 23, 27
2 A B
9-14, 17-22, 24-26
C Challenge
28-30
5
A
B
E
D
E
D
C
B
E
D
C
B
A
4
E
D
C
B
A
3
C
B
A
2
C
B
A
1
Test Prep
Mixed Review
Opposite-Direction Travel
EXAMPLE
C
D
D
E
Test-Taking Tip
E
When you grid an
integer, right-align
your answer so the
place-value is clear.
31-35
36-42
Homework Quick Check
To check students’ understanding
of key skills and concepts, go over
Exercises 8, 10, 22, 24, 18.
Exercise 6 Suggest to students
that they first think of any even
whole number and the next
greater even whole number for
part b.
.
.
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
/
3
0
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
.
/
Define Let r = Jane’s rate.
Then r + 15 = Peter’s rate.
Relate
.
Write
Person
Rate
Time
Distance
Jane
r
3
3r
Peter
r ⫹ 15
3
3(r ⫹ 15)
3r + 3(r + 15) = 225
Jane’s distance is 3r. Peter’s
distance is 3(r ⴙ 15).
The sum of Jane’s and Peter’s distances is the
total distance, 225 miles.
3r + 3(r + 15) = 225
3r + 3r + 45 = 225
6r + 45 = 225
Use the Distributive Property.
Combine like terms.
6r + 45 - 45 = 225 - 45 Subtract 45 from each side.
6r = 180
Simplify.
6r = 180
6
6
Divide each side by 6.
r = 30
Simplify.
Jane’s rate is 30 mi/h, and Peter’s rate is 15 mi/h faster, which is 45 mi/h.
Quick Check
GPS Guided Problem Solving
L3
L4
Enrichment
L2
Reteaching
L1
Adapted Practice
Practice
Name
Class
Practice 3-6
Date
L3
Absolute Value Equations and Inequalities
EXERCISES
5. Δ3c - 6« ⱖ 3
9. 9 ⬎ Δ6 + 3t«
13. Δ2n - 1« ⱖ 1
2. Δh« ⬎ 6
6. Δ2n + 3« ⱕ 5
3. Δ2k« ⬎ 8
A
7. Δ3.5z« ⬎ Δ7«
4. Δs + 4« ⬎ 2
11. 5 ⬎ Δv + 2« + 3
12. Δ4y + 11« ⬍ 7
14. ` 12 x 1 1 ` ⬎ 1
15. -2Δh - 2« ⬎ -2
16. 3Δ2x« ⱕ 12
19. 2 12 Δ6x - 3« ⱕ 2 23
20. -2Δ3j« - 8 ⱕ -20
17. 3Δs - 4« + 21 ⱕ 27 18. -6Δw - 3« ⬍ -24
Practice by Example
Example 1
8. ` 23 x ` ⱕ 4
10. Δj« - 2 ⱖ 6
Solve each equation. If there is no solution, write no solution.
21. Δa« = 9.5
22. Δb« = -2
23. Δd« - 25 = -13
24. Δ6z« + 3 = 21
25. Δ3c« - 45 = -18
uzu
26. -2 = 2 7
27. Δx« = -0.8
28. -4Δ7 + d« = -44
GO for
Help
(page 158)
Write and solve an absolute value equation or inequality that represents
each situation.
30. The mean distance of the earth from the sun is 93 million miles. The
distance varies by 1.6 million miles. Find the range of distances of the
earth from the sun.
© Pearson Education, Inc. All rights reserved.
1. The length of a rectangle is 3 in. more than its width. The perimeter of the
c. 2w ± 2(w ± 3) ≠ 30; 6
rectangle is 30 in.
a. Define a variable for the width. Let w ≠ width
b. Write an expression for the length in terms of the width. /≠ w ± 3
c. Write an equation to find the width of the rectangle. Solve your equation.
d. What is the length of the rectangle? 9 in.
See above.
2. The length of a rectangle is 8 in. more than its width. The perimeter of the
rectangle is 24 in. What are the width and length of the rectangle? 2 in.; 10 in.
29. The average number of cucumber seeds in a package is 25. The number
of seeds in the package can vary by three. Find the range of acceptable
numbers of seeds in each package.
3. The width of a rectangle is one half its length. The perimeter of the rectangle is
54 cm. What are the width and length of the rectangle? 9 cm; 18 cm
31. Leona was in a golf tournament last week. All four of her rounds of golf
were within 2 strokes of par. If par was 72, find the range of scores that
Leona could have shot for each round of the golf tournament.
32. Victor’s goal is to earn $75 per week at his after-school job. Last month
he was within $6.50 of his goal. Find the range of amounts that Victor
might have earned last month.
33. Members of the track team can run 400 m in an average time of 58.2 s.
The fastest and slowest times vary from the average by 6.4 s. Find the
range of times for the track team.
34. The ideal length of a particular metal rod is 25.5 cm. The measured
length may vary from the ideal length by at most 0.025 cm. Find the
range of acceptable lengths for the rod.
35. When measured on a particular scale, the weight of an object may vary
from its actual weight by at most 0.4 lb. If the reading on the scale is
125.2 lb, find the range of actual weights of the object.
Example 2
36. One poll reported that the approval rating of the job performance of
the President of the United States was 63%. The poll was said to be
accurate to within 3.8%. What is the range of actual approval ratings?
(page 159)
162
162
For more exercises, see Extra Skill and Word Problem Practice.
Practiceand
andProblem
ProblemSolving
Solving
Practice
Solve each inequality. Graph the solution.
1. Δd« ⬎ 2
5 Sarah and John leave Perryville traveling in opposite directions on a straight road.
Sarah drives 12 miles per hour faster than John. After 2 hours, they are 176 miles
apart. Find Sarah’s speed and John’s speed. John: 38 mi/h; Sarah: 50 mi/h
4. The length of a rectangular garden is 3 yd more than twice its width. The
perimeter of the garden is 36 yd. What are the width and length of the garden?
5 yd; 13 yd
5. The sum of three consecutive integers is 915. What are the integers?
304, 305, 306
Chapter 3 Solving Equations
6. The sum of two consecutive even integers is 118.
a. Define a variable for the smaller integer. Let n ≠ the first integer.
b. What must you add to an even integer to get the next greater even integer? 2
c. Write an expression for the second integer. n ± 2
d. Write and solve an equation to find the two even integers.
n ± n ± 2 ≠ 118; 58, 60
7. The sum of two consecutive even integers is -298. What are the integers?
–148, –150
8. The sum of two consecutive odd integers is 56.
Example 3
(page 160)
9d.
t
van
car
1
40
30
11–2
60
60
a. Define a variable for the smaller integer. Let n ≠ the first integer.
b. What must you add to an odd integer to get the next greater odd integer? 2
c. Write an expression for the second integer. n ± 2
d. Write and solve an equation to find the two odd integers.
n ± n ± 2 ≠ 56; 27, 29
9. A moving van leaves a house traveling at an average rate of 40 mi/h. The family
leaves the house 12 hour later following the same route in a car. They travel at
an average rate of 60 mi/h.
a. Define a variable for the time traveled by the moving van. Let t ≠ time for
the moving van.
b. Write an expression for the time traveled by the car. t – 12
c. Copy and complete the table.
Vehicle
✔
t ⴝ 1 1–2
t ⴚ 1–2 ⴝ 1
The car catches the
van after traveling
1 hour.
Moving van
Car
Rate
Time
■ 40
■t
■ 60
1
2
t■
–
Exercise 9 Suggest to students
that they use 0.5 instead of 12 for
the time.
Exercise 15 Help students
see that odd integers can be
represented the same way as
even integers here. Ask: What is
the difference between two
consecutive odd integers? 2
What is the difference between
two consecutive even integers? 2
Distance Traveled
■ 40t
■ 60 Qt – 12R
d. Make a table comparing the distance traveled by the moving van and car for
each hour. Use the table to find out how long it will take the car to catch up
with the van. See left.
10. Air Travel A jet leaves the Charlotte, North Carolina, airport traveling at an
average rate of 564 km/h. Another jet leaves the airport one half hour later
traveling at 744 km/h in the same direction. Use an equation to find how long
the second jet will take to overtake the first. 1 17 h
30
Example 4
(page 161)
11b. 22x ≠ 72 – 32x; 113 h
11. Juan drives to work. Because of traffic conditions, he averages 22 miles per
hour. He returns home averaging 32 miles per hour. The total travel time
is 2 14 hours.
a. Define a variable for the time Juan takes to travel to work. Write an
expression for the time Juan takes to return home. x; 2 1 – x
4
b. Write and solve an equation to find the time Juan spends driving to work.
12. Air Travel An airplane flies from New Orleans, Louisiana, to Atlanta, Georgia,
at an average rate of 320 miles per hour. The airplane then returns at an
average rate of 280 miles per hour. The total travel time is 3 hours.
a. Define a variable for the flying time from New Orleans to Atlanta. Write an
expression for the travel time from Atlanta to New Orleans. x; 3 – x
b. Write and solve an equation to find the flying time from New Orleans
to Atlanta. 320x ≠ 840 – 280x; 12 h
5
Example 5
(page 162)
13. John and William leave their home traveling in opposite directions on a straight
road. John drives 20 miles per hour faster than William. After 4 hours they are
250 miles apart. a. x; x – 20
b. 4x ± 4x – 80 ≠ 250; 4141 mi/h; 2114 mi/h
a. Define a variable for John’s rate. Write an expression for William’s rate.
b. Write and solve an equation to find John’s rate. Then find William’s rate.
14. Two bicyclists ride in opposite directions. The speed of the first bicyclist is
5 miles per hour faster than the second. After 2 hours they are 70 miles apart.
Find their rates. 15 mi/h; 20 mi/h
Lesson 3-6 Equations and Problem Solving
163
163
4. Assess & Reteach
B
15. The sum of three consecutive odd integers is -87. What are the integers?
–27, –29, –31
16. The tail of a kite is 1.5 ft plus twice the length of the kite. Together, the kite and
tail are 15 ft 6 in. long.
a. Write an expression for the length of the kite and tail together. 1.5 + 2x + x
b. Write 15 ft 6 in. in terms of feet. 15.5 ft
c. Write and solve an equation to find the length of the tail. 5
3x ± 1.5 ≠ 15.5; 10 ft or 10 ft 10 in.
17. Travel A bus traveling at an average rate of 30 miles per hour6left the city at
11:45 A.M. A car following the bus at 45 miles per hour left the city at noon. At
what time did the car catch up with the bus? 12:30 P.M.
Apply Your Skills
PowerPoint
Lesson Quiz
1. The sum of three consecutive
integers is 117. Find the
integers. 38, 39, 40
2. You and your brother started
biking at noon from places
that are 52 mi apart. You rode
toward each other and met
at 2:00 P.M. Your brother’s
average speed was 4 mi/h
faster than your average
speed. Find both speeds.
your speed: 11 mi/h;
brother’s speed: 15 mi/h
18. Ellen and Kate raced on their bicycles to the library after school. They both
left school at 3:00 P.M. and bicycled along the same path. Ellen rode at a speed
of 12 miles per hour and Kate rode at 9 miles per hour. Ellen got to the library
15 minutes before Kate. At what time did Ellen get to the library? 3:45
19. a. Which of the following numbers is not the sum of three
consecutive integers? II.
19b. They are all multiples of 3.
I. 51
II. 61
III. 72
IV. 81
b. Critical Thinking What common trait do the other numbers share?
20. At 1:30 P.M., Tom leaves in his boat from a dock and heads south. He travels at
a rate of 25 miles per hour. Ten minutes later, Mary leaves the same dock in her
speedboat and heads after Tom. If she travels at a rate of 30 miles per hour,
when will she catch up with Tom? 2:30 P.M.
3. Joan ran from her home to the
lake at 8 mi/h. She ran back
home at 6 mi/h. Her total
running time was 32 minutes.
How much time did it take
Joan to run from her home to
the lake? about 13.7 minutes
21. Air Travel Two airplanes depart from an airport traveling in opposite
directions. The second airplane is 200 miles per hour faster than the first. After
2 hours they are 1100 miles apart. Find the speeds of the airplanes. 175 mi/h; 375 mi/h
Alternative Assessment
Organize students in groups of 5.
In each group, assign each
student one of the examples from
the lesson. Instruct each student
to write a problem similar to the
assigned example on a card. Have
students in each group exchange
cards and solve the problem given
to them. Repeat until each
student in a group has solved
each problem. Have students
compare answers and strategies.
22b. the distance
travelled
24. See margin.
22. At 1:00 P.M. a truck leaves Centerville
GPS traveling 45 mi/h. One hour later a train
leaves Centerville traveling 60 mi/h. They
arrive in Smithfield at the same time.
a. Use the table to find when the train and
truck arrive in Smithfield. 4:00 P.M.
b. Critical Thinking What piece of
information can you get from the table
that you would NOT get by solving the
equation 45t = 60(t - 1)? See left.
Truck
45t
Train
60(t ⴚ 1)
1 P.M.
45 mi
0 mi
2 P.M.
90 mi
60 mi
3 P.M.
135 mi
120 mi
4 P.M.
180 mi
180 mi
5 P.M.
225 mi
240 mi
23. Three friends were born in consecutive years. The sum of their birth years is
5982. Find the year in which each person was born. 1993, 1994, 1995
26b. Yes; if n is the
middle integer,
n – 1 is the
previous integer
and n ± 1 is the
next integer. The
three integers
would be
consecutive.
GO
nline
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164
24. Writing Describe the steps you would use to solve consecutive integer problems.
25. Open-Ended Write a word problem that could be solved using the equation
35(t - 1) = 20t. See margin.
26. a. Write and solve an equation to find three consecutive integers with
a sum of 126. Let n = the first integer. n ± n ± 1 ± n ± 2 ≠ 126; 41, 42, 43
b. Critical Thinking In part (a), could you solve the problem by letting n = the
middle integer, n - 1 = the smallest integer, and n + 1 = the largest integer?
27. Electricity A group of ten
6- and 12-volt batteries are
wired in series as shown at
the right. The sum of their
voltages is 84 volts. How
many of each type of
battery are used?
6 6-V; 4 12-V
Batteries in Series
Chapter 3 Solving Equations
pages 162–165
164
Time
Exercises
24. Answers may vary.
Sample: Define a variable
to represent the first
integer. Use this variable
to write expressions for
the other integers. Write
an equation that describes
how the integers are
related. Solve this equation
to find the integers.
25. Answers may vary. Sample:
Jeff and Anne both left
school for the city at the
same time. Jeff drove
35 mi/h and Anne drove
20 mi/h. Jeff arrived 1 h
before Anne. How long
did each drive?
Test Prep
C
Challenge
28. Geometry A triangle has a perimeter of 165 cm. The first side is 65 cm less
than twice the second side. The third side is 10 cm less than the second side.
Write and solve an equation to find the length of each side of the triangle.
x ± 2x – 65 ± x – 10 ≠ 165; 60; 55 cm, 60 cm, 50 cm
29. At 9:00 A.M., your friends begin hiking at 2 mi/h. You begin from the same
place at 9:25 A.M. You hike at 3 mi/h.
a. How long will you have hiked when you catch up with your friends? 56 h
b. At what time will you catch up with your friends? 10:15 A.M.
30. Find five consecutive odd integers such that the sum of the first and the fifth is
one less than three times the fourth. –9, –7, –5, –3, –1
Resources
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 195
• Test-Taking Strategies, p. 190
• Test-Taking Strategies with
Transparencies
StandardizedTest
TestPrep
Prep
Multiple Choice
31. Solve 3n - 7 + 2n = 8n + 11. A
B. 11
A. -6
3
C. 33
5
D. 9
32. Which expression represents the sum of 3 odd integers of which n is the
least integer? H
F. n + 3
G. 3n + 3
H. 3n + 6
J. 3n + 7
33. Which equation does NOT have -2 as its solution? B
A. 2x + 5 = 5x + 11
B. 7n + 9 = 3 - 9n
C. 3k + 6 - 4k = k + 10
D. 4 + 3q = 7q + 12
34. A truck traveling at an average rate of 45 miles per hour leaves a rest stop.
Fifteen minutes later a car traveling at an average rate of 60 miles per hour
leaves the same rest stop traveling the same route. How long will it take
for the car to catch up with the truck? G
F. 15 minutes
G. 45 minutes
H. 1 hour 15 minutes
J. 3 hours
35. The perimeter of the triangle at
the right is 22.6 in. What is the
value of n? A
A. 3.5
B. 4.6
C. 7.8
D. 9.4
n in.
(n ⫹ 5.2) in.
(2n ⫹ 3.4) in.
Mixed Review
Lesson 3-3
Solve each equation. If the equation is an identity, write identity. If it has no
solution, write no solution.
36. 2x = 7x + 10 –2
37. 2q + 4 = 4 - 2q 0
38. 0.5t + 3.6 = 4.2 - 1.5t 0.3
39. 2x + 5 + x = 2(3x + 3) –13
40. 4 + x + 3x = 2(2x + 5) no solution 41. 8z + 2 = 2(z - 5) - z –157
Lesson 3-2
42. Brendan earns $8.25 per hour at his job. He also makes $12.38 per hour for
any number of hours over 40 that he works in one week. He worked 40 hours
last week, plus some overtime, and made $385.71. How many overtime hours
did he work? 4 12 h
lesson quiz, PHSchool.com, Web Code: ata-0306
Lesson 3-6 Equations and Problem Solving
165
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