Working with Algebraic Expressions
LESSON SUMMARY
In this lesson, you will use the same rules of signs that you learned in
the previous lesson for any number—including fractions—not just integers. You will find out how to simplify and evaluate expressions and see how using the order of operations can help you find the correct answer.
Simplifying Expressions
What does it mean when you are asked to simplify an expression? Numbers can be named in many different ways. For example:
1 2, 0.5, 50%, and 3 6 all name the same number.When you are told to simplify an expression, you want to get the simplest name possible. For example, because 36 can be reduced, 12 is the simplest name of the number.
Mathematical expressions, like numbers, can be named in different ways. For example, here are three ways to write the same expression:
1. x + –3
2. x + (–3) When you have two signs side by side, parentheses can be used to keep the signs separate.
3. x – 3 Remember that Lesson 1 showed that subtracting a positive 3 is the same as adding the opposite of a positive 3.
The operation of multiplication can be shown in many ways. In Lesson 1, we used the dot (·) to indicate multiplication. A graphics calculator will display an asterisk when it shows multiplication. You are probably familiar with this notation (2 × 3) to show multiplication. However, in algebra, we rarely use the × to indicate multiplication since it may be unclear whether the × is a variable or a multiplication sign. To avoid confusion over the use of the ×, we express multiplication in other ways. Another way to indicate multiplication is the use of parentheses
(2)(3); also, when you see an expression such as 3ab, it is telling you to multiply 3 times a times b.
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Mixed Practice
Here is a mixture of problems for you to solve using what you’ve learned in this lesson.Work the problems without the use of a calculator.
______65. 2 – 7
______66. –2 · 5
______67. –12 ÷ 4
______68. –9 + 12
______69. 16 – –5
______70. –36 ÷ –18
______71. 7 · –8
______72. –45 – 2
______73. –9 · –3
______74. –11 + –15
______75. 39 ÷ –3
______76. –18 – –3
______77. –20 · 3
______78. –8 – –8
______79. 37 – 12
______80. –10 – 2
______81. –2 · –7 · –3
______82. 2 + 9 –11 + 3
______83. 10 · –2 · –3
______84. –48 ÷ –3
______85. 20 – –3 + 8 – 11
______86. 2 · –2 · –4
______87. –10 – –4
______88. 60 ÷ –5
______89. 4 · –5 · –2 · 10
______90. –10 + 5 – 7 – 13
Applications
Represent the information in the problem with signed numbers. Then solve the problem.
Example: In Great Falls,Montana the temperature changed from 40 to 3 below zero.What was the difference in temperature?
40° – –3° = 43°
91. The weather channel stated the high temperature for the day was 85° in Phoenix and the low temperature was 17° in Stanley, Idaho.What is the difference in the temperatures?
92. The predicted high for the day in Bismarck, North Dakota was 30° and the low was 7° below zero.What is the difference in the temperatures for the day?
93. You have a bank balance of $45 and write a check for $55.What is your new balance?
94. You have an overdraft of $20. You deposit $100.What is your new balance?
95. The water level in the town reservoir goes down 8 inches during a dry month, then gains 5 inches in a heavy rainstorm, and then loses another inch during the annual lawn sprinkler parade.What is the overall effect on the water level?
Skill Building until Next Time
When you balance your checkbook, you are working with positive and negative numbers. Deposits are positive numbers. Checks and service charges are negative numbers. Balance your checkbook using positive and negative numbers.
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Multiplying and Dividing Integers
When you are multiplying or dividing integers, if the signs are the same, the answer will be positive. If the signs are different, the answer will be negative. The dot (·) indicates multiplication in the following examples.
Examples: 2 · 5 = 10
–2 · –5 = 10
–2 · 5 = –10
10 ÷ –2 = –5
–10 ÷ 2 = –5
–10 ÷ –2 = 5
Here’s a tip to use when multiplying more than two numbers at a time.
Tip
If you are multiplying more than two numbers, use the odd-even rule. Count the number of negative signs in the problem. If the problem has an odd number of negative signs, the answer will be negative. If there is an even number of negative signs, the answer will be positive.
Examples: 2 · 3 · –5 = –30
–5 · 2 · –3 = 30
–7 · 3 · –2 · –1 = –42
Note: Zero is considered an even number.
Practice
______49. 7 · 8
______50. –4 · 5
______51. –14 ÷ 2
______52. –12 · –2
______53. –56 · –8
______54. –33 ÷ 3
______55. –44 ÷ –11
______56. 24 ÷ –3
______57. 7 · –11
______58. –75 ÷ –3
______59. 3 · 2 · –4
______60. 5 · –4 · –2
______61. –3 · –2 · –6
______62. 3 · 5 · 6
______63. –2 · 3 · –1 · –4
______64. –4 · –5 · –2 · –2
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Shortcuts and Tips
Here are some tips that can shorten your work and save you time!
Tip #1: You may have discovered that (– –) will be the same as a positive number.Whenever you have a problem with two negative signs side by side, change both signs to positive. Then work the problem.
Example: 4 – –8 = 4 + +8 = 4 + 8 = 12
Tip #2: Notice that the subtraction sign is bigger and lower than the negative sign. You may have discovered that the subtraction sign gives you the same answer as a negative sign. You will find that the most frequently used notation is 5 – 9 rather than 5 + –9.
Example: 3 – 5 = 3 + –5 = –2 so 3 – 5 = –2
Tip #3:When adding more than two numbers, add all the positive numbers, add all the negative numbers, then add the resulting positive and negative numbers to obtain the answer.
Example: 2 + 3 + 5 – 7 = 10 – 7 = 3
–2 + 5 + 7 + –6 = 12 + –8 = 4
5 – 7 + 6 – 9 = 11 + –16 = –
Practice
Practice using the shortcuts.
______29. 3 – –2
______30. 5 –11
______31. 5 – –7
______32. 9 – –5
______33. –1 –1
______34. –6 – 5
______35. –11 – –12
______36. –7 – –7
______37. 13 –5
______38. –8 – –12
______39. 4 + 11 + 5 – 10
______40. 7 + 4 – 5 + 2
______41. –7 + 5 + 9 – 4
______42. 12 – 3 + 4 + 6
______43. –9 – 11 – 2 + 5
______44. –8 + 12 – 5 + 6
______45. 14 – 11 + 7 – 6
______46. –5 – –7 + –4 + 10
______47. –2 + 5 + 7 – 3 + 6
______48. 3 – 10 – –6 + 5 + –7
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What Are the Sign Rules for Subtracting Integers?
All subtraction problems can be converted to addition problems because subtracting is the same as adding the opposite. Once you have converted the subtraction problem to an addition problem, use the Rules for Adding Integers on the previous page.
For example, 2 – 5 can also be written as 2 – +5. Subtraction is the same as adding the opposite, so 2 – +5 can be rewritten as 2 + –5. Because the problem has been rewritten as an addition problem, you can use the Rules for Adding Integers. As you recall, the rule says that if the signs are different, you should subtract the numbers and
take the sign of the larger number and disregard the sign. Therefore, 2 + –5 = –3. See the following examples.
Examples: 7 – 3 = 7 – +3 = 7 + –3 = 4
6 – –8 = 6 + +8 = 14
–5 – –11 = –5 + +11 = 6
Practice
______17. 5 – 6
______18. 3 ___–6
______19. –2 – 5
______20. –7 – 12
______21. –9 – 3
______22. –15 – –2
______23. –8 – –2
______24. –11 – –6
______25. 10 – 3
______26. 6 – –6
______27. 9 – 9
______28. –8 – 10
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Adding and Subtracting Integers
You can add and subtract integers; but before you can do that, you need to know the rules for determining the sign of an answer. Of course, a calculator can determine the sign of your answer when you are working with integers. However, it is important that you know how to determine the sign of your answer without the use of a calculator. There are situations where you will be unable to use a calculator and will be handicapped if you do not know the rules for determining the sign of an answer. Knowing how to determine the sign of your answer is a basic algebra skill and is absolutely necessary to progress to more advanced algebra skills.
What Are the Sign Rules for Adding Integers?
When the signs of the numbers are the same, add the numbers and keep the same sign for your answer.
Examples: –3 + –5 = –8
4 + 3 = 7
If the signs of the numbers are different (one is positive and one is negative), then treat both of them as positive for a moment. Subtract the smaller one from the larger one, then give this answer the sign of the larger one.
Examples: 4 + –7 = –3
The answer is negative because 7 is bigger than 4 when we ignore signs.
Practice
9. 7 + 5
______10. –4 + –8
______11. –17 + 9
______12. –9 + –2
______13. –3 + 10
______14. 3 + –9
______15. 11 + –2
______16. –5 + 5
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Working with Integers
LESSON SUMMARY
Algebra is the branch of mathematics that denotes quantities with letters and uses negative numbers as well as ordinary numbers. In this lesson, you will be working with a set of numbers called integers. You use integers in your daily life. A profit is represented with a positive number and a loss is shown using a negative number. This lesson defines integers and explains the rules for adding, subtracting, multiplying, and dividing integers.
What Is an Integer?
Integers are all the positive whole numbers (whole numbers do not include fractions), their opposites, and zero. For example, the opposite of 2 (positive 2) is the number –2 (negative 2). The opposite of 5 (positive 5) is –5 (negative 5). The opposite of 0 is 0. Integers are often called “signed numbers” because we use the positive and negative signs to represent the numbers. The numbers greater than zero are positive numbers, and the numbers less than zero are negative numbers. If the temperature outside is 70°, the temperature is represented with a positive number. However, if the temperature outside is 3 below zero, we represent this number as –3, which is a negative number.
Integers can be represented in this way:
… –3, –2, –1, 0, 1, 2, 3, …
The three dots that you see at the beginning and the end of the numbers mean the numbers go on forever in both directions. Notice that the numbers get increasingly smaller when you advance in the negative direction and increasingly larger when you advance in the positive direction. For example, –10 is less than –2. The mathematical
symbol for less than is “<” so we say that –10 < –2. The mathematical symbol for greater than is “>”. Therefore, 10 > 5. If there is no sign in front of a number, it is assumed the number is a positive number.
Practice
Insert the correct mathematical symbol > or < for the following pairs of numbers. Check your answers with the answer key at the end of the book.
1. 5___11
2. 1___–2
3. –4___0
4. –2___–8
5. –35___–18
6. 0___–6
7. 12___0
8. 14___–23
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PRETEST (5)
41. Solve the equation: 3×2 – 27 = 0.
a. 0, 3
b. 3, 3
c. 3, –3
d. –3, –3
42. Solve the equation: 2×2 + x = 3.
a. –1, 2
3
b. –
32
, 1
c. 3, –2
1
d. –3, 2
1
43. Simplify: –3x + 2x + 3y.
a. x + 3y
b. 2xy
c. –x + 3y
d. –5x + 3y
Team-LRN
44. Simplify: 310xy · 46x
a. 720xy
b. 1260xy
c. 24x15y
d. 482xy
45. Simplify:18 + 52.
a. 820
b. 105
c. 82
d. 152
46. Simplify:
a.
8
3
b.
2
3
2
c.
2
3
6
d. 210
47. Solve the equation:x + 3 = 5.
a. 2
b. 8
c. 4
d. 64
48. Solve the equation: 3x+ 1 = 15.
a. 4
b. 24
c. 26
d. 6
49. Use the quadratic formula to solve: 3×2 + x – 2.
a. 3, –1
b.
2
3
, –1
c. –
2
3
, 1
d. –
1
6
,
5
6
50. Use the quadratic formula to solve: 4×2 – 3x – 2.
a.
b.
c.
d.
3 ±17
8
–3 ±17
8
3 ±41
8
–3 ±41
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PRETEST (4)
31. Solve the system of equations algebraically:
2x – y = 10
3x + y = 15
a. (0,5)
b. (5,0)
c. (–5,0)
d. (0,–5)
32. Solve the system of equations algebraically:
4x – 3y = 10
5x + 2y = 1
a. (4, –3)
b. (1, –2)
c. (–1, –
1
3
)
d. (2, –
2
3
)
33. Simplify: 2×2y(3×3y2).
a. 6×6y2
b. 6×5y2
c. 6×5y3
d. 6×6y3
34. Simplify: 5(2xy3)3.
a. 10×3y6
b. 10×3y9
c. 11×4y6
d. 40×3y9
35. Multiply the polynomials: 2×2(3x + 4xy – 2xy3).
a. 6×3 + 8×2y – 4×3y3
b. 6×3 + 8×3y – 4×3y3
c. 6×3 + 8×3y – 4×2y3
d. 6×2 + 8×2y – 4×3y3
36. Multiply the binomials: (2x + 3)(x – 2).
a. 2×2 + x – 6
b. 2×2 – x + 6
c. 2×2 – x – 6
d. 2×2 + x + 6
37. Factor the polynomial: 3a2b + 6a3b2 – 15a2b4.
a. 3a2b(2ab – 5b3)
b. 3a2b(1 + 2a2b – 5b3)
c. 3a2b(1 + 2ab + 5b3)
d. 3a2b(1 + 2ab – 5b3)
38. Factor the polynomial: 49w2 – 81.
a. (7w + 9)(7w – 9)
b. (7w – 9)(7w – 9)
c. (7w + 9)(7w + 9)
d. (7w – 9)2
39. Factor the polynomial: x2 + 3x – 18.
a. (x – 2)(x + 9)
b. (x + 3)(x – 6)
c. (x + 2)(x – 9)
d. (x – 3)(x + 6)
40. Factor the polynomial: 10×2 + 13x – 3.
a. (2x + 3)(5x – 1)
b. (2x – 3)(5x + 1)
c. (2x + 1)(5x – 3)
d. (2x – 1)(5x + 3)
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PRETEST (3)
21. What is the slope in the equation y =
2
3
x + 5?
a.
3
2
b.
2
3
c. 2
d. 5
22. Transform the equation 3x + y = 5 into slopeintercept
form.
a. y = 3x + 5
b. y = –3x + 5
c. x =
13
y + 5
d. x = –
13
y + 5
23. Choose the equation that fits the graph.
a. y = 2x + 3
b. 3x + y = 2
c. –3x + y = 2
d. y = –2x + 3
24. Solve the inequality: 4x + 4 > 24.
a. x > 7
b. x > 5
c. x < 5
d. x < 7
25. Solve the inequality: x + 5 ? 3x + 9.
a. x ?
7
2
b. x ? –2
c. x ? –2
d. x ? 2
26. Match the graph with the inequality: y > 4.
a.
b.
c.
d.
10
10
–10
–10
10
10
–10
–10
10
10
–10
–10
10
10
–10
–10
27. Match the inequality with the graph.
a. y < 2x + 3
b. y ? 2x + 3
c. y > 2x + 3
d. y ? 2x + 3
28. Determine the number of solutions the system
of equations has by looking at the graph.
a. 1
b. 0
c. infinite
d. none of the above
29. Use the slope and intercept to determine the
number of solutions to the system of linear
equations:
3y + 6 = 2x
3y = 2x + 6
a. 0
b. 1
c. ?
d. none of the above
30. Select the graph for the system of
inequalities:
y > 2
y ? 2x + 1
a.
b.
c.
d.
10
10
–10
–10
10
–10
10
10
–10
–10
10
10
–10
–10
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