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Name Date Class
Ready to Go On? Skills Intervention
SECTION
12-1
Solving Two-Step Equations
To solve equations with more than one operation, or a two-step
equation, follow the order of operations in reverse. First add or
subtract and then multiply or divide.
Solving Two-Step Equations Using Division
Solve. Check your answer.
7x (cid:2) 9 (cid:3) 37
What number do you subtract from both sides?
7x (cid:3)
7x 28
(cid:2)(cid:2)(cid:3) (cid:2)(cid:2) What number do you divide by to isolate the variable?
x (cid:3) What does x equal?
Check:
7x (cid:2) 9 (cid:3) 37
?
7( ) (cid:2) 9 (cid:3) 37 Substitute for x into the equation.
?
(cid:2) 9 (cid:3) 37 Multiply.
(cid:3) 37 Does the answer check?
Solving Two-Step Equations Using Multiplication
Solve. Check your answer.
p
9 (cid:2) (cid:2)(cid:2) (cid:3) 16
8
What number do you subtract from both sides?
p
(cid:2)(cid:2) (cid:3)
8
p
( ) (cid:2)(cid:2) (cid:3) (7) What number do you multiply by to isolate the variable?
8
p (cid:3)
Check:
p
9 (cid:2) (cid:2)(cid:2) (cid:3) 16
8
56 ?
9 (cid:2) (cid:2)(cid:2) (cid:3) 16 Substitute for p into the equation.
8
?
9 (cid:2) (cid:3) 16 Divide.
(cid:3) 16 Does the answer check?
Copyright ©by Holt, Rinehart and Winston. 216 Holt Mathematics
All rights reserved.
Name Date Class
Ready to Go On? Problem Solving Intervention
LESSON
12-1
Solving Two-Step Equations
Sometimes you can write one equation to solve a problem that
would otherwise take two steps to solve.
Your teacher says that if you get a 100 on the next test, your mean
score will be 94. What is your mean score for the 6 tests you have
taken so far?
Understand the Problem
1. List what you know and what you are asked to find.
2. Complete to show how to find
the mean of a set of values. mean =
Make a Plan
3. Let your current mean score be x. Write an expression
with x for the sum of your 6 scores so far.
Hint: mean for 6 scores(cid:3)(cid:2)sum of 6(cid:2)scores
6
4. Suppose you get a 100 on the next test. Write an addition
expression with x for the sum of your 7 test scores.
Use your answer from Exercise 3.
5. Complete the equation to show that your new mean score will
be 94. Hint: Use your answer from Exercise 4 to write an
expression with x for your new mean score.
(cid:2)
(cid:2)(cid:2) (cid:3)
Solve
6. Solve the equation from Exercise 5 and answer the question in
the problem.
Check
7. Why should your answer be less than 94?
Copyright ©by Holt, Rinehart and Winston. 217 Holt Mathematics
All rights reserved.
MSM07C2_RTGO_ch12_216-233_B 6/17/06 3:49 PM Page 218 (Black plate)
Name Date Class
Ready to Go On? Skills Intervention
LESSON
12-2
Solving Multi-Step Equations
There is usually more than one way to solve a multi-step equation. There are steps you
can take, however, to make the equation easier to solve. For instance, adding like
terms on each side of the equation simplifies the solving process. Also, using the
Distributive Property helps simplify the equation.
Combining Like Terms to Solve an Equation
Solve 5x (cid:1) 12 (cid:2) 1x (cid:3) 36.
5x (cid:1) 12 (cid:2) 1x (cid:3) 36
x (cid:1) 12 (cid:3) 36 Combine like terms on the left side of the equation.
Subtract from both sides.
4x 24
(cid:1)(cid:1) (cid:3) (cid:1)(cid:1) Divide both sides by to isolate the variable.
4
x (cid:3) Solve for x.
Using the Distributive Property to Solve Equations
Solve 2(3x (cid:2) 5) (cid:1) 7 (cid:3) 12
2(3x (cid:2) 5) (cid:1) 7 (cid:3) 12
(cid:2) (cid:1) 7 (cid:3) 12 Distribute 2 on the left side.
– (cid:3) 12 Combine like terms.
Add to both sides.
(cid:3) Divide both sides by .
x (cid:3) Simplify.
x (cid:3)
Application
Lisa’s collection of wild animal cards has 9 less than half of 6 more
than 4 times the number her sister has. Lisa has 110 wild animal
cards. How many does her sister have?
1
(cid:1)(cid:1)(4x (cid:1)6) (cid:2)9(cid:3)110 Set up the equation.
2
(cid:1) (cid:2)9 (cid:3) 110 Use the Distributive Property.
(cid:2) (cid:3) 110 Combine like terms.
(cid:3) Add to both sides.
(cid:3) Divide both sides by .
Lisa’s sister has wild animal cards.
Copyright ©by Holt, Rinehart and Winston. 218 Holt Mathematics
All rights reserved.
Name Date Class
Ready to Go On? Problem Solving Intervention
LESSON
12-2
Solving Multi-Step Equations
Sometimes you can write one equation to solve a problem that
would take more than two steps to solve.
Acircular path just fits inside a square park. Walking around the d
circle is 12 kilometers shorter than walking around the outside of the
22
park. What is d, the distance across the park? Use (cid:2)(cid:2) for (cid:2).
7
Understand the Problem
1. Complete the word equation to show what you know.
(cid:3)perimeter of park (cid:4)
2. Complete each sentence.
d is the of the circle
d is the of the square (or width or side)
Make a Plan
3. Use d to write expressions for the perimeter of the square
park and for the circumference of the circular path.
4. Write an equation you can use to solve for d. Hint: Substitute
the expressions from Exercise 3 into the word equation
from Exercise 1.
Solve
5. Solve the equation you wrote in Exercise 4 and answer the question in the
22
problem. Use (cid:2)(cid:2) for (cid:2).
7
Check
6. Start with your answer for d and see if the circumference of the circle is 12
kilometers less than the perimeter of the park.
Solve
7. Find d if the difference in paths is 24 km instead of 12 km.
Copyright ©by Holt, Rinehart and Winston. 219 Holt Mathematics
All rights reserved.
Name Date Class
Ready to Go On? Skills Intervention
LESSON
12-3
Solving Equations with Variables on Both Sides
When solving equations with variables on both sides of the
equation, get all of the variable terms on one side of the equation by
adding or subtracting.
Using Inverse Operations to Group Terms with Variables
Group the terms with variables on one side of the equal sign and simplify.
A. 4g (cid:2) 4 (cid:3) 6g
4g (cid:2) 4 (cid:3) 6g To get the variables on one side of the equation,
what do you add or subtract from both sides?
4 (cid:3) g Combine like terms.
4 2g
(cid:2)(cid:2) (cid:3) (cid:2)(cid:2) By what number do you divide both sides?
2 (cid:3) g Solve for g.
B. (cid:4)5d (cid:2) 8 (cid:3) 11d
(cid:4)5d (cid:2) 8 (cid:3) 11d To get the variables on one side of the equation,
what do you add or subtract from both sides?
8 (cid:3) d Combine like terms.
8 16d
(cid:2)(cid:2)(cid:3) (cid:2)(cid:2) By what number do you divide both sides?
(cid:3) d Solve for d.
Consumer Math Application
To do her laundry, Chrissy used to pay $10 per month to rent a
washer and a dryer. However, Chrissy just purchased a new
washing machine for $210. She still rents the dryer for $3 per
month. How many months will it take for the amount it cost Chrissy
to rent the washer and dryer to cost as much as it does for her to
buy the new washing machine and rent only the dryer?
210 (cid:2) 3t (cid:3) 10t Let t represent the number of .
What do you subtract from both sides?
210 (cid:3) t Combine like terms.
210 7t
(cid:2)(cid:2) (cid:3) (cid:2)(cid:2) By what number do you divide both sides?
(cid:3) t Solve for t.
After months or years, the cost will be the same.
Copyright ©by Holt, Rinehart and Winston. 220 Holt Mathematics
All rights reserved.
Name Date Class
Ready to Go On? Problem Solving Intervention
LESSON
12-3
Solving Equations with Variables on Both Sides
You can convert a temperature in degrees Celsius to a temperature in degrees
9
Fahrenheit by using the equation F(cid:3)(cid:2)(cid:2)C(cid:2)32. At what temperature is the
5
number of degrees Celsius the same as the number of degrees Fahrenheit?
Understand the Problem
9
1. In the formula F(cid:3)(cid:2)(cid:2)C(cid:2)32, what does F stand for? What does C stand for?
5
2. When the temperature is 0°C, it is 32°F. How does that you tell that 0° is not
the answer?
Make a Plan
9
3. Why will solving the equation C(cid:3)(cid:2)(cid:2)C(cid:2)32 give you the answer to the problem?
5
Solve
9 9 4
4. Solve the equation C(cid:3)(cid:2)(cid:2)C(cid:2)32 for C. Hint: C(cid:4)(cid:2)(cid:2)C(cid:3)(cid:4)(cid:2)(cid:2)C.
5 5 5
5. At what temperature is the number of degrees Celsius
the same as the number of degrees Fahrenheit?
Check
9
6. Substitute your answer for both F and C in the equation F(cid:3)(cid:2)(cid:2)C (cid:2)32 and see
5
if the equation is true.
Copyright ©by Holt, Rinehart and Winston. 221 Holt Mathematics
All rights reserved.
MSM07C2_RTGO_ch12_216-233_B 6/17/06 3:49 PM Page 222 (Black plate)
Name Date Class
Ready to Go On? Quiz
SECTION
12A
12-1 Solving Two-Step Equations
Solve.
d
1. 3b (cid:2)9 (cid:3) 27 2. (cid:1)(cid:1) (cid:2) 5 (cid:3) 4
7
3. (cid:2)14 (cid:3) 7 (cid:1) 3s 4. (cid:2)4 (cid:2) 6x (cid:3) (cid:2)64
5. 7k (cid:2) 5 (cid:3) 51 6. (cid:2)23 (cid:3) 17 (cid:1) 4f
y
7. (cid:1)(cid:1) (cid:1) 12 (cid:3) 17 8. –5t (cid:2) 6 (cid:3) 34
3
9. Stacey’s Barbershop charges $12 for a haircut. The tenth
cut is free. If $132 is spent, how many haircuts are received?
12-2 Solving Multi-Step Equations
Solve.
2c(cid:2)16
10. (cid:1)(cid:1) (cid:3)7 11. 4(7e(cid:2)9) (cid:1)6(cid:3)82
4
2.03(cid:1)6r 1 2w (cid:2)8
12. (cid:1)(cid:1) (cid:3)2.09 13. (cid:1)(cid:1) (cid:3)(cid:1)(cid:1)
7 4 16
14. 3(6m (cid:2)4) (cid:1)10 (cid:3)70 15. 5(2n(cid:1)4) (cid:2)30 (cid:3)100
Copyright ©by Holt, Rinehart and Winston. 222 Holt Mathematics
All rights reserved.
MSM07C2_RTGO_ch12_216-233_B 6/17/06 3:49 PM Page 223 (Black plate)
Name Date Class
Ready to Go On? Quiz
continued
SECTION
12A
12-2 Solving Multi-Step Equations (continued)
16. Yolanta got $2.72 in change from a $20 music store gift
certificate. She paid 8% in tax. What did she spend before tax?
17. A brother and sister wanted to treat their mother to a birthday
massage. The cost of the massage was $72, including a 20%
tip. How much was the total bill before the tip?
12-3 Solving Equations with Variables on Both Sides
Solve.
4 2
18. 3b (cid:3)18b(cid:2)180 19. (cid:1)(cid:1)y (cid:1) 6 (cid:3) (cid:1)(cid:1)y (cid:2) 7
5 5
7 3 1 15
20. (cid:2)35s (cid:1) 71 (cid:3) (cid:2)21s (cid:1) 43 21. (cid:1)(cid:1)t (cid:1) (cid:1)(cid:1) (cid:3) (cid:1)(cid:1) (cid:1) (cid:1)(cid:1)t
8 4 2 16
22. 6.5g (cid:2) 7.2 (cid:3) 9.4 (cid:2) 1.8g 23. 5h (cid:2) 9 (cid:3) (cid:2)3h (cid:1) 15
24. One cell phone service charges $0.10 per minute with no
activation fee. Another cell phone service charges $0.05 per
minute with a $20 activation fee. Find the number of minutes for
which the two cell phone services charge the same amount.
Copyright ©by Holt, Rinehart and Winston. 223 Holt Mathematics
All rights reserved.
MSM07C2_RTGO_ch12_216-233_B 6/17/06 3:49 PM Page 224 (Black plate)
Name Date Class
Ready to Go On? Enrichment
SECTION
12A
Puzzle Pieces
For the junior achievement contest, one group set out to create a
jigsaw puzzle. Before deciding on their final product a number of
design options were considered. They agreed on the number of
pieces per puzzle, the puzzle dimensions, and the colors. The final
product was a black and white puzzle of the Brooklyn Bridge.
Solving Equations with Variables on Both Sides
1. Two types of puzzles were created, one with large pieces for
young children and another with small pieces for older children.
Puzzle A had 160 pieces that averaged 0.5 inches and puzzle B
had 400 pieces that averaged 0.2 inches. What is the smallest
size of wood that could be used for both puzzles?
2. To find out how much cutting would be needed, the length of
each border and interior cut was measured in centimeters.
9 • (number of interior cuts) (cid:2)5(cid:3)6 • (number of border cuts) (cid:1)4
Find the length per cut.
3. An equal number of puzzles were shipped to two locations. One
place got 6 times the number of boxes minus the 5 that were
returned and another location was sent 4 times the number of
boxes plus 3. Find the number of boxes shipped.
4. 3 times the number of black paint tubes plus 2 is equal to
5 times the number of white paint tubes minus 4. Find the
number of paint tubes.
Copyright ©by Holt, Rinehart and Winston. 224 Holt Mathematics
All rights reserved.
Name Date Class
Ready to Go On? Skills Intervention
LESSON
12-4
Inequalities
An inequality states that two values either are not equal or may
Vocabulary
not be equal. An inequality uses one of the following:
inequality
(cid:4) less than (cid:5) less than or equal to algebraic
(cid:6) greater than (cid:7) greater than or equal to inequality
An inequality that contains a variable is an algebraic inequality.
Writing Inequalities
Write an inequality for each situation.
A. There are at least 6 runners on a track team.
number of runners 6 “At least” means than or equal to.
B. There are fewer than 12 students in government.
number of students 12 “Fewer than” means than.
C. No more than 50 people can swim in the pool.
number of people 50 “No more than” means less than or .
Graphing Inequalities
Graph each inequality.
A. f (cid:4) 3
(cid:2)5 (cid:2)4 (cid:2)3 (cid:2)2 (cid:2)1 0 1 2 3 4 5
Draw an circle on 3 because 3 included as a solution.
Draw an arrow from 3 going to the of 3 because the numbers
smaller than 3 are all to the of 3.
B. k (cid:5) 1
(cid:2)5 (cid:2)4 (cid:2)3 (cid:2)2 (cid:2)1 0 1 2 3 4 5
Draw a circle on 1 because 1 included as a solution.
Draw an arrow from 1 going to the of 1 because the numbers
smaller than 1 are all to the of 1.
Copyright ©by Holt, Rinehart and Winston. 225 Holt Mathematics
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