Preparing for Geometry

Now

Chapter 0 contains lessons on topics from previous courses.

You can use this chapter in various ways.
Begin the school year by taking the Pretest. If you need
additional review, complete the lessons in this chapter. To
verify that you have successfully reviewed the topics, take
the Posttest.
As you work through the text, you may find that there are
topics you need to review. When this happens, complete the
individual lessons that you need.
Use this chapter for reference. When you have questions
about any of these topics, flip back to this chapter to review
definitions or key concepts.

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Animation Vocabulary eGlossary Audio Foldables Worksheets
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Get Started on the Chapter
You will review several concepts, skills, and vocabulary terms as you study Chapter 0.
To get ready, identify important terms and organize your resources.

StudyOrganizer ReviewVocabulary
Throughout this text, you will be invited to use Foldables to English Español
organize your notes. experiment p. P8 experimento
trial p. P8 prueba
Why should you use them?
outcome p. P8 resultado
They help you organize, display, and arrange information.
event p. P8 evento
They make great study guides, specifically designed for you.
probability p. P8 probabilidad
You can use them as your math journal for recording main theoretical probability p. P9 probabilidad teórica
ideas, problem-solving strategies, examples, or questions you
may have. experimental probability p. P9 probabilidad experimental
ordered pair p. P15 par ordenado
They give you a chance to improve your math vocabulary.
x-coordinate p. P15 coordenada x
How should you use them? y-coordinate p. P15 coordenada y
Write general information — titles, vocabulary terms, quadrant p. P15 cuadrante
concepts, questions, and main ideas — on the front tabs of origin p. P15 origen
your Foldable.
system of equations p. P17 sistema de ecuaciones
Write specific information — ideas, your thoughts, answers to
substitution p. P17 sustitución
questions, steps, notes, and definitions — under the tabs.
elimination p. P18 eliminación
Use the tabs for:
Product Property p. P19 Propriedad de Producto
• math concepts in parts, like types of triangles,
Quotient Property p. P19 Propriedad de Cociente
• steps to follow, or
• parts of a problem, like compare and contrast (2 parts) or
what, where, when, why, and how (5 parts).
You may want to store your Foldables in a plastic zipper bag
that you have three-hole punched to fit in your notebook.

When should you use them?

Set up your Foldable as you begin a chapter, or when you start
learning a new concept.
Write in your Foldable every day.
Use your Foldable to review for homework, quizzes, and tests.

P2 | Chapter 0 | Preparing for Geometry

 Pretest
State which metric unit you would probably use to Solve each inequality.
measure each item.
36. y - 13 < 2 37. t + 8 ≥ 19
1. length of a computer keyboard
38. _
n
> -6 39. 9a ≤ 45
4
2. mass of a large dog
40. x + 12 > -14 41. -2w < 24
42. -_
n
≥3 43. -_
b
≤ -6
Complete each sentence. 7 5

3. 4 ft = ___
? in. 4. 21 ft = ___
? yd
5. 180 g = ___
? kg 6. 3 T = ___
? lb Write the ordered pair for each point shown.
44. F y
7. 32 g ≈ ___
? oz 8. 3 mi ≈ ___
? km
D
9. 35 yd ≈ ___
? m 10. 5.1 L ≈ ___
? qt 45. H
46. A F

11. TUNA A can of tuna is 6 ounces. About how many 47. D A

O x
grams is it? H

12. CRACKERS A box of crackers is 453 grams.

About how many pounds is it? Round to the
nearest pound.
Graph and label each point on the coordinate
13. DISTANCE A road sign in Canada gives the distance plane above.
to Toronto as 140 kilometers. What is this distance
to the nearest mile? 48. B(4, 1) 49. G(0, -3)
50. R(-2, -4) 51. P(-3, 3)
PROBABILITY A bag contains 3 blue chips, 7 red chips,
4 yellow chips, and 5 green chips. A chip is randomly
52. Graph the triangle with vertices J(1, -4), K(2, 3),
drawn from the bag. Find each probability.
and L(-1, 2).
14. P(yellow) 15. P(green)
53. Graph four points that satisfy the equation
16. P(red or blue) 17. P(not red) y = 2x - 1.

Evaluate each expression if r = 3, q = 1, and w = -2.

Solve each system of equations.
18. 4r + q 19. rw - 6
r + 3q
54. y = 2x 55. -3x - y = 4
20. _ 21. _
5w
y = -x + 6 4x + 2y = -8
4r 3r + q
22. |2 - r| + 17 23. 8 + |q - 5| 56. y = 2x + 1 57. _
1
x - y = -1
2
y = 3x x - 2y = 5
Solve each equation. 58. x + y = -6 59. _
1
x - 3y = -4
3
24. k + 3 = 14 25. a - 7 = 9 2x - y = 3 x - 9y = -12
26. 5c = 20 27. n + 2 = -11
28. 6t - 18 = 30 29. 4x + 7 = -1 Simplify.
30. _r = -8 31. _
3
b = -2 60. √
18 61. 
_
25
4 5 49
32. -_ w
= -9 33. 3y - 15 = y + 1
2 62. √
24x 2y 3 63. _
3
34. 27 - 6d = 7 + 4d 35. 2(m - 16) = 44 4 - √
5

connectED.mcgraw-hill.com P3
 Changing Units of Measure
Within Systems
Objective
Convert units of measure
within the customary and Example 1 Choose Best Unit of Measure
metric systems.
State which metric unit you would use to measure the length of your pen.
A pen has a small length, but not very small. The centimeter is the appropriate
unit of measure.

Metric Units of Length Customary Units of Length

1 kilometer (km) = 1000 meters (m) 1 foot (ft) = 12 inches (in.)
1 m = 100 centimeters (cm) 1 yard (yd) = 3 ft
1 cm = 10 millimeters (mm) 1 mile (mi) = 5280 ft

• To convert from larger units to smaller units, multiply.

• To convert from smaller units to larger units, divide.
• To use dimensional analysis, multiply by the ratio of the units.

Example 2 Convert from Larger Units to Smaller Units of Length

Complete each sentence.
a. 4.2 km = ? m b. 13 yd = ? ft
There are 1000 meters in a kilometer. There are 3 feet in a yard.
4.2 km × 1000 = 4200 m 13 yd × 3 = 39 ft

Example 3 Convert from Smaller Units to Larger Units of Length

Complete each sentence.
a. 17 mm = ? m
There are 100 centimeters in a meter. First change millimeters to centimeters.
17 mm = ? cm smaller unit → larger unit
17 mm ÷ 10 = 1.7 cm Since 10 mm = 1 cm, divide by 10.
Then change centimeters to meters.
1.7 cm = ? m smaller unit → larger unit
1.7 cm ÷ 100 = 0.017 m Since 100 cm = 1 m, divide by 100.

b. 6600 yd = mi ?
Use dimensional analysis.
6600 yd × _
3 ft
×_
1 mi
= 3.75 mi
1 yd 5280 ft

Metric Units of Capacity Customary Units of Capacity

1 liter (L) = 1000 milliliters (mL) 1 cup (c) = 8 fluid ounces (fl oz) 1 quart (qt) = 2 pt
1 pint (pt) = 2 c 1 gallon (gal) = 4 qt

P4 | Lesson 0-1
StudyTip Example 4 Convert Units of Capacity
Dimensional Analysis You
Complete each sentence.
can use dimensional analysis
for any conversion in this a. 3.7 L = ? mL b. 16 qt = ? gal
lesson.
There are 1000 milliliters in a liter. There are 4 quarts in a gallon.
3.7 L × 1000 = 3700 mL 16 qt ÷ 4 = 4 gal

c. 7 pt = ? fl oz d. 4 gal = ? pt
There are 8 fluid ounces in a cup. There are 4 quarts in a gallon.
First change pints to cups. First change gallons to quarts.
7 pt = ? c 4 gal = ? qt
7 pt × 2 = 14 c 4 gal × 4 = 16 qt
Then change cups to fluid ounces. Then change quarts to pints.
14 c = ? fl oz 16 qt = ? pt
14 c × 8 = 112 fl oz 16 qt × 2 = 32 pt

The mass of an object is the amount of matter that it contains.

Metric Units of Mass Customary Units of Weight
1 kilogram (kg) = 1000 grams (g) 1 pound (lb) = 16 ounces (oz)
1 g = 1000 milligrams (mg) 1 ton (T) = 2000 lb

Example 5 Convert Units of Mass

Complete each sentence.
a. 5.47 kg = ? mg b. 5 T = ? oz
There are 1000 milligrams in a gram. There are 16 ounces in a pound.
Change kilograms to grams. Change tons to pounds.
5.47 kg = ? g 5 T = ? lb
5.47 kg × 1000 = 5470 g 5 T × 2000 = 10,000 lb
Then change grams to milligrams. Then change pounds to ounces.
5470 g = ? mg 10,000 lb = ? oz
5470 g × 1000 = 5,470,000 mg 10,000 lb × 16 = 160,000 oz

Exercises
State which metric unit you would probably use to measure each item.
1. radius of a tennis ball 2. length of a notebook 3. mass of a textbook
4. mass of a beach ball 5. liquid in a cup 6. water in a bathtub

Complete each sentence.

7. 120 in. = ? ft 8. 18 ft = ? yd 9. 10 km = ? m
10. 210 mm = ?cm 11. 180 mm = ? m 12. 3100 m = ? km
13. 90 in. = ? yd 14. 5280 yd = ? mi 15. 8 yd = ? ft
16. 0.62 km = ? m 17. 370 mL = ? L 18. 12 L = ? mL
19. 32 fl oz = ? c 20. 5 qt = ? c 21. 10 pt = ? qt
22. 48 c = ? gal 23. 4 gal = ? qt 24. 36 mg = ? g
25. 13 lb = ? oz 26. 130 g = ? kg 27. 9.05 kg = ? g

connectED.mcgraw-hill.com P5
 Changing Units of Measure
Between Systems
Objective
Convert units of measure The table below shows approximate equivalents between customary units of length
between the customary and metric units of length.
and metric systems.
Units of Length
Customary → Metric Metric → Customary
1 in. ≈ 2.5 cm 1 cm ≈ 0.4 in.
1 yd ≈ 0.9 m 1 m ≈ 1.1 yd
1 mi ≈ 1.6 km 1 km ≈ 0.6 mi

Example 1 Convert Units of Length Between Systems

Complete each sentence.
a. 30 in. ≈ ? cm b. 5 km ≈ ? mi
There are approximately There is approximately 0.6 mile
2.5 centimeters in an inch. in a kilometer.
30 in. × 2.5 = 75 cm 5 km × 0.6 = 3 mi

Example 2 Convert Units of Length Between Systems

Complete: 2000 yd ≈ ? km.

There is approximately 0.9 meter in a yard. First find the number of meters in
2000 yards.
2000 yd × 0.9 = 1800 m
Then change meters to kilometers. There are 1000 meters in a kilometer.
1800 m ÷ 1000 = 1.8 km

The table below shows approximate equivalents between customary units of capacity
and metric units of capacity.

Units of Capacity
Customary → Metric Metric → Customary
1 qt ≈ 0.9 L 1 L ≈ 1.1 qt
1 pt ≈ 0.5 L 1 L ≈ 2.1 pt

Example 3 Convert Units of Capacity Between Systems

Complete each sentence.
a. 7 qt ≈ ? L b. 2 L ≈ ? pt
There is approximately 0.9 liter There are approximately 2.1 pints
in a quart. in a liter.
7 qt × 0.9 = 6.3 L 2 L × 2.1 = 4.2 pt

P6 | Lesson 0-2
 Example 4 Convert Units of Capacity Between Systems

Complete: 10 L ≈ ? gal.
There are approximately 1.1 quarts in a liter. First find the number of quarts in
10 liters.
10 L × 1.1 = 11 qt
Then change quarts to gallons. There are 4 quarts in a gallon.
StudyTip 11 qt ÷ 4 = 2.75 gal
Dimensional Analysis If the
You can also use dimensional analysis.
unit that you want to
eliminate is in the numerator, 1.1 qt 1 gal
make sure it is in the 10 L × _ × _ = 2.75 gal
1L 4 qt
denominator of the ratio
when you multiply. If it is in
the denominator, make sure
that it is in the numerator of
The table below shows approximate equivalents between customary units of weight and
the ratio.
metric units of mass.

Units of Weight/Mass
Customary → Metric Metric → Customary
1 oz ≈ 28.3 g 1 g ≈ 0.04 oz
1 lb ≈ 0.5 kg 1 kg ≈ 2.2 lb

Example 5 Convert Units of Mass Between Systems

Complete each sentence.
a. 58.5 kg ≈ ? lb b. 14 oz ≈ ? g
There are approximately 2.2 pounds There are approximately 28.3 grams
in a kilogram. in an ounce.
58.5 kg × 2.2 = 128.7 lb 14 oz × 28.3 = 396.2 g

Exercises
Complete each sentence.
1. 8 in. ≈ ? cm 2. 15 m ≈ ? yd 3. 11 qt ≈ ? L

4. 25 oz ≈ ? g 5. 10 mi ≈ ? km 6. 32 cm ≈ ? in.

7. 20 km ≈ ? mi 8. 9.5 L ≈ ? qt 9. 6 yd ≈ ? m
10. 4.3 kg ≈ ? lb 11. 10.7 L ≈ ? pt 12. 82.5 g ≈ ? oz

13. 2_
1
lb ≈ ? kg 14. 10 ft ≈ ? m 15. 1_
1
gal ≈ ? L
4 2
16. 350 g ≈ ? lb 17. 600 in. ≈ ? m 18. 2.1 km ≈ ? yd

19. CEREAL A box of cereal is 13 ounces. About how many grams is it?

20. FLOUR A bag of flour is 2.26 kilograms. How much does it weigh? Round to the
nearest pound.

21. SAUCE A jar of tomato sauce is 1 pound 10 ounces. About how many grams is it?

connectED.mcgraw-hill.com P7
 Simple Probability
Objective
Find the probability of A situation involving chance such as flipping a coin or rolling a die is an experiment.
simple events. A single performance of an experiment such as rolling a die one time is a trial. The result
of a trial is called an outcome. An event is one or more outcomes of an experiment.

When each outcome is equally likely to happen, the probability of an event is the
NewVocabulary ratio of the number of favorable outcomes to the number of possible outcomes. The
experiment probability of an event is always between 0 and 1, inclusive.
trial
outcome equally likely to occur
event
probability
theoretical probability impossible to occur certain to occur
experimental probability 1 1 3
0 4 2 4
1

0% 25% 50% 75% 100%

Common Core
State Standards
Content Standards Example 1 Find Probability
S.MD.6 (+) Use probabilities
to make fair decisions (e.g., Suppose a die is rolled. What is the probability of rolling an odd number?
drawing by lots, using a
random number generator). There are 3 odd numbers on a die: 1, 3, and 5.
S.MD.7 (+) Analyze There are 6 possible outcomes: 1, 2, 3, 4, 5, and 6.
decisions and strategies
P(odd) = ___
using probability concepts number of favorable outcomes
(e.g., product testing, number of possible outcomes
medical testing, pulling a
hockey goalie at the end of =_
3
or _
1
a game). 6 2

Mathematical Practices The probability of rolling an odd number is _

1
or 50%.
2
4 Model with mathematics.

For a given experiment, the sum of the probabilities of all possible outcomes must sum to 1.

Example 2 Find Probability

Suppose a bag contains 4 red, 3 green, 6 blue, and 2 yellow marbles. What is the
probability a randomly chosen marble will not be yellow?
Since the sum of the probabilities of all of the colors must sum to 1, subtract the
probability that the marble will be yellow from 1.
The probability that the marble will be yellow is _2
because there are 2 yellow marbles
15
and 15 total marbles.
P(not yellow) = 1 - P(yellow)

=1-_
2
15
=_
13
15

The probability that the marble will not be yellow is _

13
or about 87%.
15

P8 | Lesson 0-3
 The probabilities in Examples 1 and 2 are called theoretical probabilities. The theoretical
probability is what should occur. The experimental probability is what actually occurs
when a probability experiment is repeated many times.

StudyTip
Experimental Probability
Example 3 Find Experimental Probability
The experimental probability The table shows the results of an
of an experiment is not Outcome Tally Frequency
experiment in which a number cube
necessarily the same as the 1 6
was rolled. Find the experimental
theoretical probability, but
probability of rolling a 3. 2 4
when an experiment is
repeated many times, the 3 7
experimental probability P(3) = ___
number of times 3 occurs
or _
7
4 3
total number of outcomes 25
should be close to the
The experimental probability for getting 5 4
theoretical probability.
a 3 in this case is _
7
or 28%. 6 1
25

Exercises
A die is rolled. Find the probability of each outcome.
1. P(less than 3) 2. P(even) 3. P(greater than 2)
4. P(prime) 5. P(4 or 2) 6. P(integer)

A jar contains 65 pennies, 27 nickels, 30 dimes, and 18 quarters. A coin is randomly

selected from the jar. Find each probability.
7. P(penny) 8. P(quarter)
9. P(not dime) 10. P(penny or dime)
11. P(value greater than $0.15) 12. P(not nickel)
13. P(nickel or quarter) 14. P(value less than $0.20)

PRESENTATIONS The students in a class are randomly drawing cards numbered 1 through
28 from a hat to determine the order in which they will give their presentations. Find
each probability.
15. P(13) 16. P(1 or 28) 17. P(less than 14)
18. P(not 1) 19. P(not 2 or 17) 20. P(greater than 16)

The table shows the results of an experiment in which three coins were tossed.
Outcome HHH HHT HTH THH TTH THT HTT TTT
Tally
Frequency 5 5 6 6 7 5 8 8

21. What is the experimental probability that all three of the coins will be heads? The
theoretical probability?
22. What is the experimental probability that at least two of the coins will be heads? The
theoretical probability?
23. DECISION MAKING You and two of your friends have pooled your money to buy a new
video game. Describe a method that could be used to make a fair decision as to who
gets to play the game first.
24. DECISION MAKING A new study finds that the incidence of heart attack while taking a
certain diabetes drug is less than 5%. Should a person with diabetes take this drug?
Should they take the drug if the risk is less than 1%? Explain your reasoning.

connectED.mcgraw-hill.com P9
 Algebraic Expressions
Objective
Use the order of An expression is an algebraic expression if it contains sums and/or products of
operations to evaluate variables and numbers. To evaluate an algebraic expression, replace the variable
algebraic expressions. or variables with known values, and then use the order of operations.
Order of Operations
Step 1 Evaluate expressions inside grouping symbols.
Step 2 Evaluate all powers.
Step 3 Do all multiplications and/or divisions from left to right.
Step 4 Do all additions and/or subtractions from left to right.

Example 1 Addition/Subtraction Algebraic Expressions

Evaluate x - 5 + y if x = 15 and y = -7.
x - 5 + y = 15 - 5 + (-7) Substitute.
= 10 + (-7) or 3 Subtract.

Example 2 Multiplication/Division Algebraic Expressions

Evaluate each expression if k = -2, n = -4, and p = 5.

a. _
2k + n
b. -3(k 2 + 2n)
p-3
2(-2) + (-4)
_ = __
2k + n
Substitute. -3(k 2 + 2n) = -3[(-2) 2 + 2(-4)]
p-3 5-3
=_-4 - 4
Multiply. = -3[4 + (-8)]
5-3
=_-8
or -4 Subtract. = -3(-4) or 12
2

Example 3 Absolute Value Algebraic Expressions

Evaluate 3|a - b| + 2|c - 5| if a = -2, b = -4, and c = 3.
3|a - b| + 2|c - 5| = 3|-2 - (-4)| + 2|3 - 5| Substitute for a, b, and c.
= 3|2| + 2|-2| Simplify.
= 3(2) + 2(2) or 10 Find absolute values.

Exercises
Evaluate each expression if a = 2, b = -3, c = -1, and d = 4.
1. 2a + c 2. _
bd
3. _
2d - a
4. 3d - c
2c b
5. _
3b
6. 5bc 7. 2cd + 3ab 8. _
c - 2d
a
5a + c

Evaluate each expression if x = 2, y = -3, and z = 1.

9. 24 + |x - 4| 10. 13 + |8 + y| 11. |5 - z| + 11 12. |2y - 15| + 7

P10 | Lesson 0-4

 Linear Equations
Objective
Use algebra to solve linear If the same number is added to or subtracted from each side of an equation, the resulting
equations. equation is true.

Example 1 Addition/Subtraction Linear Equations

Solve each equation.
a. x - 7 = 16
x - 7 = 16 Original equation
x - 7 + 7 = 16 + 7 Add 7 to each side.
x = 23 Simplify.

b. m + 12 = -5
m + 12 = -5 Original equation
m + 12 + (-12) = -5 + (-12) Add -12 to each side.
m = -17 Simplify.

c. k + 31 = 10
k + 31 = 10 Original equation
k + 31 - 31 = 10 - 31 Subtract 31 from each side.
k = -21 Simplify.

If each side of an equation is multiplied or divided by the same number, the resulting
equation is true.

Example 2 Multiplication/Division Linear Equations

Solve each equation.
a. 4d = 36
4d = 36 Original equation
_
4d
=_
36
Divide each side by 4.
4 4
x=9 Simplify.
_
b. - t = -7
8
-_t = -7 Original equation
8
-8 - _t = -8(-7)
( ) Multiply each side by -8.
8
t = 56 Simplify.

c. _
3
x = -8
5
_3 x = -8 Original equation
5
_5 _3 x = _5 (-8)
() _
Multiply each side by 5 .
3 5 3 3
x = -_ 40
Simplify.
3

To solve equations with more than one operation, often called multi-step equations, undo
operations by working backward.

connectED.mcgraw-hill.com P11
 Example 3 Multi-step Linear Equations
Solve each equation.
a. 8q - 15 = 49
8q - 15 = 49 Original equation

8q = 64 Add 15 to each side.

q=8 Divide each side by 8.

b. 12y + 8 = 6y - 5
12y + 8 = 6y - 5 Original equation

12y = 6y - 13 Subtract 8 from each side.

6y = -13 Subtract 6y from each side.

13
y = -_ Divide each side by 6.
6
WatchOut!
Order of Operations
When solving equations that contain grouping symbols, first use the Distributive
Remember that the order of
Property to remove the grouping symbols.
operations applies when you
are solving linear equations.
Example 4 Multi-step Linear Equations
Solve 3(x - 5) = 13.
3(x - 5) = 13 Original equation

3x - 15 = 13 Distributive Property

3x = 28 Add 15 to each side.

x=_
28
Divide each side by 3.
3

Exercises
Solve each equation.
1. r + 11 = 3 2. n + 7 = 13 3. d - 7 = 8
p
4. _
8
a = -6 5. -_ = 6 6. _
x
=8
5 12 4
_ y
_
7. 12 f = -18 8. = -11 9. _
6
y=3
5 7 7
10. c - 14 = -11 11. t - 14 = -29 12. p - 21 = 52

13. b + 2 = -5 14. q + 10 = 22 15. -12q = 84

16. 5t = 30 17. 5c - 7 = 8c - 4 18. 2 + 6 = 6 - 10

19. _
m
+ 15 = 21 20. -_
m
+7=5 21. 8t + 1 = 3t - 19
10 8
22. 9n + 4 = 5n + 18 23. 5c - 24 = -4 24. 3n + 7 = 28

25. -2y + 17 = -13 26. -_

t
-2=3 27. _
2
x-4=_
2
13 9 3
28. 9 - 4g = -15 29. -4 - p = -2 30. 21 - b = 11
31. -2(n + 7) = 15 32. 5(m - 1) = -25 33. -8a - 11 = 37

34. _
7
q - 2 = -5 35. 2(5 - n) = 8 36. -3(d - 7) = 6
4

P12 | Lesson 0-5 | Linear Equations

 Linear Inequalities
Objective
Use algebra to solve linear Statements with greater than (>), less than (<), greater than or equal to (≥), or less than
inequalities. or equal to (≤) are inequalities. If any number is added or subtracted to each side of an
inequality, the resulting inequality is true.

Example 1 Addition/Subtraction Linear Inequalities

Solve each inequality.
a. x - 17 > 12
x - 17 > 12 Original inequality
x - 17 + 17 > 12 + 17 Add 17 to each side.
x > 29 Simplify.

The solution set is {x|x > 29}.

b. y + 11 ≤ 5
y + 11 ≤ 5 Original inequality
y + 11 - 11 ≤ 5 - 11 Subtract 11 from each side.
y ≤ -6 Simplify.

The solution set is {y|y ≤ -6}.

If each side of an inequality is multiplied or divided by a positive number, the resulting

inequality is true.

Example 2 Multiplication/Division Linear Inequalities

Solve each inequality.
a. _t ≥ 11 b. 8p < 72
6
_t ≥ 11 Original inequality 8p < 72 Original inequality
6
8p
_
_
(6) t ≥ (6)11 Multiply each side by 6. <_
72
Divide each side by 8.
6 8 8
t ≥ 66 Simplify. p <9 Simplify.
The solution set is {t|t ≥ 66}. The solution set is {p|p < 9}.

If each side of an inequality is multiplied or divided by the same negative number, the
direction of the inequality symbol must be reversed so that the resulting inequality is true.

Example 3 Multiplication/Division Linear Inequalities

Solve each inequality.
a. -5c > 30
-5c > 30 Original inequality
_
-5c
<_
30
Divide each side by -5. Change > to <.
-5 -5
c < -6 Simplify.
The solution set is {c|c < -6}. (continued on the next page)

connectED.mcgraw-hill.com P13
 _
b. - d ≤ -4
13
-_
d
≤ -4 Original inequality
13

(-13) _
-d
13( )
≥ (-13)(-4) Multiply each side by -13. Change ≤ to ≥.

d ≥ 52 Simplify.

The solution set is {d|d ≥ 52}.

Inequalities involving more than one operation can be solved by undoing the operations
in the same way you would solve an equation with more than one operation.

Example 4 Multi-Step Linear Inequalities

Solve each inequality.
a. -6a + 13 < -7
-6a + 13 < -7 Original inequality
-6a + 13 - 13 < -7 - 13 Subtract 13 from each side.
WatchOut! -6a < -20 Simplify.
_ >_
Dividing by a Negative -6a -20
Divide each side by -6. Change < to >.
Remember that any time you -6 -6
divide an inequality by a
a> _
10
Simplify.
negative number you reverse 3
the direction of the sign. ⎧ ⎫
The solution set is ⎨a a > _
10
⎬.3
⎩ ⎭
b. 4z + 7 ≥ 8z - 1
4z + 7 ≥ 8z - 1 Original inequality
4z + 7 - 7 ≥ 8z - 1 - 7 Subtract 7 from each side.
4z ≥ 8z - 8 Simplify.
4z - 8z ≥ 8z - 8 - 8z Subtract 8z from each side.
-4z ≥ -8 Simplify.
_
-4z
≤_
-8
Divide each side by -4. Change ≥ to ≤.
-4 -4
z≤2 Simplify.
The solution set is {z|z ≤ 2}.

Exercises
1. x - 7 < 6 2. a + 7 ≥ -5 3. 4y < 20
4. -_
a
<5 5. _t > -7 6. _
a
≤8
8 6 11
7. d + 8 ≤ 12 8. m + 14 > 10 9. 12k ≥ -36
10. 6t - 10 ≥ 4t 11. 3z + 8 < 2 12. 4c + 23 ≤ -13
13. m - 21 < 8 14. x - 6 ≥ 3 15. -3b ≤ 48
p
16. -_ ≥ 14 17. 2z - 9 < 7z + 1 18. -4h > 36
5
_
2
19. b - 6 ≤ -2 20. _
8
t + 1 > -5 21. 7q + 3 ≥ -4q + 25
5 3
22. -3n - 8 > 2n + 7 23. -3w + 1 ≤ 8 24. -_
4
k - 17 > 11
5

P14 | Lesson 0-6 | Linear Inequalities

 Ordered Pairs
Objective
Name and graph points in Points in the coordinate plane are named by ordered pairs of the form (x, y). The first
the coordinate plane. number, or x-coordinate, corresponds to a number on the x-axis. The second number, or
y-coordinate, corresponds to a number on the y-axis.

Example 1 Writing Ordered Pairs

NewVocabulary
ordered pair Write the ordered pair for each point. y
x-coordinate
y-coordinate a. A
quadrant The x-coordinate is 4.
origin B
The y-coordinate is -1. O x
The ordered pair is (4, -1). A

b. B
The x-coordinate is -2.
The point lies on the x-axis, so its y-coordinate is 0.
The ordered pair is (-2, 0).

The x-axis and y-axis separate the coordinate plane y

into four regions, called quadrants. The point at
Quadrant II Quadrant I
which the axes intersect is called the origin. The (-, +) (+, +)
axes and points on the axes are not located in any
of the quadrants. O x

Quadrant III Quadrant IV

(-, -) (+, -)

Example 2 Graphing Ordered Pairs

Graph and label each point on a coordinate plane.
Name the quadrant in which each point is located.
a. G(2, 1) y
H(-4, 3)
Start at the origin. Move 2 units right, since the
x-coordinate is 2. Then move 1 unit up, since the G(2, 1)
y-coordinate is 1. Draw a dot, and label it G.
Point G(2, 1) is in Quadrant I. O x

b. H(-4, 3)
J(0, -3)
Start at the origin. Move 4 units left, since the
x-coordinate is -4. Then move 3 units up, since
the y-coordinate is 3. Draw a dot, and label it H.
Point H(-4, 3) is in Quadrant II.
c. J(0, -3)
Start at the origin. Since the x-coordinate is 0, the point lies on the y-axis.
Move 3 units down, since the y-coordinate is -3. Draw a dot, and label it J.
Because it is on one of the axes, point J(0, -3) is not in any quadrant.

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 Example 3 Graphing Multiple Ordered Pairs
Graph a polygon with vertices A(-3, 3), y
B(1, 3), C(0, 1), and D(-4, 1). A(-3, 3)
B(1, 3)
Graph the ordered pairs on a coordinate plane.
Connect each pair of consecutive points. The C(0, 1)
D(-4, 1)
polygon is a parallelogram. O x

StudyTip Example 4 Graphing and Solving for Ordered Pairs

Lines There are infinitely Graph four points that satisfy the equation y = 4 - x.
many points on a line, so
when you are asked to find Make a table. Plot the points.
points on a line, there are Choose four values for x. y
many answers. (0, 4)
Evaluate each value of x for 4 - x. (1, 3)
(2, 2)
x 4-x y (x, y) (3, 1)
0 4-0 4 (0, 4) O x
1 4-1 3 (1, 3)
2 4-2 2 (2, 2)
3 4-3 1 (3, 1)

Exercises
Write the ordered pair for each point shown y
at the right.
K
1. B 2. C 3. D Q B P
D
4. E 5. F 6. G F H
O W x
7. H 8. I 9. J J C
I
10. K 11. W 12. M
E G
13. N 14. P 15. Q M N

Graph and label each point on a coordinate plane. Name the quadrant in which each
point is located.
16. M(-1, 3) 17. S(2, 0) 18. R(-3, -2) 19. P(1, -4)
20. B(5, -1) 21. D(3, 4) 22. T(2, 5) 23. L(-4, -3)

Graph the following geometric figures.

24. a square with vertices W(-3, 3), X(-3, -1), Z(1, 3), and Y(1, -1)
25. a polygon with vertices J(4, 2), K(1, -1), L(-2, 2), and M(1, 5)
26. a triangle with vertices F(2, 4), G(-3, 2), and H(-1, -3)

Graph four points that satisfy each equation.

27. y = 2x 28. y = 1 + x 29. y = 3x - 1 30. y = 2 - x

P16 | Lesson 0-7 | Ordered Pairs

 Systems of Linear Equations
Objective
Use graphing, substitution, Two or more equations that have common variables are called a system of equations.
and elimination to solve The solution of a system of equations in two variables is an ordered pair of numbers
systems of linear that satisfies both equations. A system of two linear equations can have zero, one, or
equations. an infinite number of solutions. There are three methods by which systems of equations
can be solved: graphing, elimination, and substitution.

Example 1 Graphing Linear Equations

NewVocabulary
system of equations Solve each system of equations by graphing. Then determine whether each system
substitution has no solution, one solution, or infinitely many solutions.
elimination a. y = -x + 3 y
y = 2x - 3
The graphs appear to intersect at (2, 1). y = -x + 3
(2, 1)
Check this estimate by replacing x with
2 and y with 1 in each equation. O x
CHECK y = -x + 3 y = 2x - 3 y = 2x - 3
1  -2 + 3 1  2(2) - 3
1=1  1=1 
The system has one solution at (2, 1).
b. y - 2x = 6 y
3y - 6x = 9 y - 2x = 6

The graphs of the equations are parallel lines.

Since they do not intersect, there are no solutions
of this system of equations. Notice that the lines 3y - 6x = 9
have the same slope but different y-intercepts. x
Equations with the same slope and the same O
y-intercepts have an infinite number of solutions.

It is difficult to determine the solution of a system when the two graphs intersect at
noninteger values. There are algebraic methods by which an exact solution can be found.
One such method is substitution.

Example 2 Substitution
Use substitution to solve the system of equations.
y = -4x
2y + 3x = 8
Since y = -4x, substitute -4x for y Use y = -4x to find the value of y.
in the second equation. y = -4x First equation
2y + 3x = 8
2(-4x) + 3x = 3
Second equation
y = -4x
( _)
= -4 - 8
5
x = -8
5
_
=_32
Simplify.
-8x + 3x = 8 Simplify. 5
8 _
-5x = 8 Combine like terms. The solution is -_( , 32 . )
5 5
_
-5x
=_
8
Divide each side by -5.
-5 -5
x = -_
8
Simplify.
5

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 Sometimes adding or subtracting two equations together will eliminate one
variable. Using this step to solve a system of equations is called elimination.

Example 3 Elimination
Use elimination to solve the system of equations.
3x + 5y = 7
4x + 2y = 0

Either x or y can be eliminated. In this example, we will eliminate x.

3x + 5y = 7 Multiply by 4. 12x + 20y = 28

4x + 2y = 0 Multiply by -3. + (-12x) - 6y = 0

14y = 28 Add the equations.
14y
_ =_
28
Divide each side by 14.
14 14
y=2 Simplify.

Now substitute 2 for y in either equation to find the value of x.

4x + 2y = 0 Second equation

4x + 2(2) = 0 y=2

4x + 4 = 0 Simplify.

4x + 4 - 4 = 0 - 4 Subtract 4 from each side.

4x = -4 Simplify.
StudyTip _
4x
=_
-4
Divide each side by 4.
Checking Solutions You can 4 4
confirm that your solutions x = -1 Simplify.
are correct by substituting
the values into both of the The solution is (-1, 2).
original equations.

Exercises
Solve by graphing.
1. y = -x + 2 2. y = 3x - 3 3. y - 2x = 1
y = -_
1
x+1 y=x+1 2y - 4x = 1
2

Solve by substitution.
4. -5x + 3y = 12 5. x - 4y = 22 6. y + 5x = -3
x + 2y = 8 2x + 5y = -21 3y - 2x = 8

Solve by elimination.
7. -3x + y = 7 8. 3x + 4y = -1 9. -4x + 5y = -11
3x + 2y = 2 -9x - 4y = 13 2x + 3y = 11

Name an appropriate method to solve each system of equations. Then solve the system.
10. 4x - y = 11 11. 4x + 6y = 3 12. 3x - 2y = 6
2x - 3y = 3 -10x - 15y = -4 5x - 5y = 5

13. 3y + x = 3 14. 4x - 7y = 8 15. x + 3y = 6

-2y + 5x = 15 -2x + 5y = -1 4x - 2y = -32

P18 | Lesson 0-8 | Systems of Linear Equations

 Square Roots and Simplifying Radicals
Objective
Evaluate square roots A radical expression is an expression that contains a square root. The expression is in
and simplify radical simplest form when the following three conditions have been met.
expressions.
• No radicands have perfect square factors other than 1.
• No radicands contain fractions.
• No radicals appear in the denominator of a fraction.
NewVocabulary The Product Property states that for two numbers a and b ≥ 0, √ a · √
ab = √ b.
Product Property
Quotient Property Example 1 Product Property
Simplify.
a. √
45
√
45 = √
3·3·5 Prime factorization of 45
= √
3 2 · √
5 Product Property of Square Roots
= 3 √
5 Simplify.

6 · √
b. √ 15
6 · √
√ 15 = √
6 · 15 Product Property
= √
3·2·3·5 Prime factorization
= √
3 2 · √
10 Product Property
= 3 √
10 Simplify.

For radical expressions in which the exponent of the variable inside the radical is even
and the resulting simplified exponent is odd, you must use absolute value to ensure
nonnegative results.

Example 2 Product Property

Simplify √
20x 3y 5z 6 .

√
20x 3y 5z 6 = √
22 · 5 · x3 · y5 · z6 Prime factorization
= √
22 · √5 · √x3 · √ y5 · √
z6 Product Property
2 3
= 2 · √
5·x· x
√ · y · √
y · |z | Simplify.
2 3
= 2xy |z | √
5xy Simplify.

√_ba = _
√a.
The Quotient Property states that for any numbers a and b, where a ≥ 0 and b ≥ 0,
√
b

Example 3 Quotient Property

Simplify √_

25
16
.

 
25 √
√_
25
16
=_
√
16
Quotient Property

=_
5
Simplify.
4

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 Rationalizing the denominator of a radical expression is a method used to eliminate radicals
from the denominator of a fraction. To rationalize the denominator, multiply the expression
by a fraction equivalent to 1 such that the resulting denominator is a perfect square.

Example 4 Rationalize the Denominator

Simplify.

a. _
2
b. _
√
13y
√
3 √
18
WatchOut!
·_ _
√
3 √
13y √
13y
=_
√ _
_
2 2
=_
Rationalizing the 3
Multiply by . Prime factorization
Denominator Don’t forget to √
3 √
3 √
3 √
3 √
18 √
2·3·3
multiply both the numerator √
13y
2 √3
and denominator by the =_ Simplify. =_ Product Property
3 3 √2
radical when you rationalize
the denominator. √
=_·

13y _
√
2
Multiply by _
√
2
.
3 √
2 √
2 √
2

√26y
=_ Product Property
6

Sometimes, conjugates are used to simplify radical expressions. Conjugates are binomials
q + r √t and p √
of the form p √ q - r √t .

Example 5 Conjugates
Simplify _
3
.
5 - √
2

·_ _
_ 5 + √ 5 + √
=_
3 3 2 2
=1
5 - √
2 5 - √
2 5 + √
2 5 + √
2
3(5 + √
2)
= _2 (a - b)(a + b) = a 2 - b 2
5 2
- ( √
2)
15 + 3 √
=_
2
2) = 2
Multiply. ( √
2
25 - 2
15 + 3 √
=_
2
Simplify.
23

Exercises
Simplify.
1. √
32 2. √
75 50 · √
3. √ 10 4. √
12 · √
20

5. √
6 · √
6 6. √
16 · √
25 7. √
98x3y6 8. √
56a2b4c5

  
9. √_
81
49
10. √_

121
16
11. √_
63
8
12. √_
288
147
3
√10p √
108 7 √
3
13. _ 14. _ 15. _
4
16. _
√
27 √
2q6 5 - 2 √
3 5 - 2 √
6

√ 3 √
5
17. _ 18. _ 19. _ 20. _
3 24 3
√
48 √
125 2 - √
2 -2 + √
13

P20 | Lesson 0-9 | Square Roots and Simplifying Radicals

 Posttest
State which metric unit you would probably use to Solve each inequality.
measure each item.
30. 4y - 9 > 1 31. -2z + 15 ≥ 4
1. mass of a book
32. 3r + 7 < r - 8 33. -_
2
k - 20 ≤ 10
5
2. length of a highway
34. -3(b - 4) > 33 35. 2 - m ≤ 6m - 12
36. 8 ≤ r - 14 37. _
2
n<_
3
n-5
Complete each sentence. 3 9

3. 8 in. = ___
? ft 4. 6 yd = ___
? ft
Write the ordered pair for each point shown.
5. 24 fl oz = ___
? pt 6. 3.7 kg = ___
? lb
38. M M y
7. 4.2 km = ___
? m 8. 285 g = ___
? kg 39. N P
9. 0.75 kg = ___
? mg 10. 1.9 L = ___
? qt 40. P
O x
41. Q Q
11. PROBABILITY The table shows the results of an
experiment in which a number cube was rolled.
N
Find the experimental probability of rolling a 4.
Outcome Tally Frequency
1 4
2 6 Graph and label each point on the coordinate
3 5
plane above.
4 3 42. A(-2, 0) 43. C(1, 3)
5 7 44. D(-4, -4) 45. F(3, -5)

CANDY A bag of candy contains 3 lollipops, 8 peanut 46. Graph the quadrilateral with vertices
butter cups, and 4 chocolate bars. A piece of candy is R(2, 0), S(4, -2), T(4, 3), and W(2, 5).
randomly drawn from the bag. Find each probability.
12. P(peanut butter cup) 47. Graph three points that satisfy the equation
13. P(lollipop or peanut butter cup) y=_
1
x - 5.
2
14. P(not chocolate bar)
15. P(chocolate bar or lollipop) Solve each system of equations.
48. 2r + m = 11 49. 2x + 4y = 6
Evaluate each expression if x = 2, y = -3, and z = 4. 6r - 2m = -2 7x = 4 + 3y
16. 6x - z 17. 6y + xz 50. 2c + 6d = 14 51. 5a - b = 17
18. 3yz 19. _
6z -_
7
+_
1
c = -d 3a + 2b = 5
xy 3 3
y + 2x
20. _ 21. 7 + |y - 11| 52. 6d + 3f = 12 53. 4x - 5y = 17
10z 2d = 8 - f 3x + 4y = 5

Solve each equation.

Simplify.
22. 9 + s = 21 23. h - 8 = 12
54. √
80 55. 
_
128
5
24. _
4m
= 18 25. _
2
d = 10

57. _
14 9 7x 3
36 · √
56. √ 81
26. 3(20 - b) = 36 27. 37 + w = 5w - 27 3

58. _
5
28. _
x
=7 29. _
1
(n + 5) = 16 81
59. √
12x 5y 2
6 4

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