Algebra I Workbook For Dummies
Page 3
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14. (–4) + (–6) + (–10) =
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15. 5 + (–18) + (10) =
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16. (–4) + 4 + (–5) + 5 + (–6) =
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Making a Difference with Signed Numbers
You really don’t need a new set of rules when subtracting signed numbers. You just change the subtraction problem to an addition problem and use the rules for addition of signed numbers. To ensure that the answer to this new addition problem is the answer to the original subtraction problem, you change the operation from subtraction to addition, and you change the sign of the second number — the one that’s being subtracted.
To subtract two signed numbers:
a – (+b) = a + (–b) and a – (–b) = a + (+b)
Q. (–8) – (–5) =
Change the problem to (–8) + (+5) =
A. –3
Q. 6 – (+11) =
Change the problem to 6 + (–11) =
A. –5
17. 5 – (–2) =
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18. –6 – (–8) =
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19. 4 – 87 =
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20. 0 – (–15) =
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21. 2.4 – (–6.8) =
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22. –15 – (–11) =
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Multiplying Signed Numbers
When you multiply two or more numbers, you just multiply them without worrying about the sign of the answer until the end. Then to assign the sign, just count the number of negative signs in the problem. If the number of negative signs is an even number, the answer is positive. If the number of negative signs is odd, the answer is negative.
The product of two signed numbers:
(+)(+) = + and (–)(–) = +
(+)(–) = – and (–)(+) = –
The product of more than two signed numbers:
(+)(+)(+)(–)(–)(–)(–) has a positive answer because there are an even number of negative factors.
(+)(+)(+)(–)(–)(–) has a negative answer because there are an odd number of negative factors.
Q. (–2)(–3) =
There are two negative signs in the problem.
A. +6
Q. (–2)(+3)(–1)(+1)(–4) =
There are three negative signs in the problem.
A. –24
23. (–6)(3) =
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24. (14)(–1) =
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25. (–6)(–3) =
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26. (6)(–3)(4)(–2) =
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27. (–1)(–1)(–1)(–1)(–1)(2) =
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28. (–10)(2)(3)(1)(–1) =
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Dividing Signed Numbers
The rules for dividing signed numbers are exactly the same as those for multiplying signed numbers — as far as the sign goes. (See “Multiplying Signed Numbers” earlier in this chapter.) The rules do differ though because you have to divide, of course.
When you divide signed numbers, just count the number of negative signs in the problem — in the numerator, in the denominator, and perhaps in front of the problem. If you have an even number of negative signs, the answer is positive. If you have an odd number of negative signs, the answer is negative.
Q.
A. +4. There are two negative signs in the problem, which is even, so the answer is positive.
Q.
A. –9. There are three negative signs in the problem, which is odd, so the answer is negative.
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32.
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33.
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34.
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Answers to Problems on Signed Numbers
This section provides the answers (in bold) to the practice problems in this chapter.
1. Which is larger, –2 or –8? The answer is –2 is larger. The following number line shows that the number –2 is to the right of –8. So –2 is bigger than –8 (or –2 > –8).
2. Which has the greater value, –13 or 2? 2 is greater. The number 2 is to the right of –13. So 2 has a greater value than –13 (or 2 > –13).
3. Which is bigger, –0.003 or –0.03? –0.003 is bigger. The following number line shows that the number –0.003 is to the right of –0.03, which means –0.003 is bigger than –0.03 (or –0.003 > –0.03).
4. Which is larger, or ? is larger. The number= , and is to the left of on the following number line. So is larger than (or ).
5. because 8 > 0.
6. because –6 < 0 and 6 is the opposite of –6.
7. because as in the previous problem.
8. because .
9. because 4 is the greater absolute value.
10. because –11 has the greater absolute value.
11. because both of the numbers have negative signs; when the signs are the same, find the sum of their absolute values.
12. because 47 has the greater absolute value.
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Or you may prefer to add the two numbers with the same sign first, like this:
You can do this because order and grouping (association) don’t matter in addition.
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23. because the multiplication problem has one negative, and 1 is an odd number.
24. because the multiplication problem has one negative, and 1 is an odd number.
25. because the multiplication problem has two negatives, and 2 is an even number.
26. because the multiplication problem has two negatives.
27. because the multiplication problem has five negatives.
28. because the multiplication problem has two negatives.
29. because the division problem has two negatives.
30. because the division problem has one negative.
31. because three negatives result in a negative.
32. because the division problem has two negatives.
33. because the division problem has five negatives.
34. because the division problem has one negative.
Chapter 2
Incorporating Algebraic Properties
In This Chapter
Embracing the different types of grouping symbols
Distributing over addition and subtraction
Utilizing the associative and commutative rules
Algebra has rules for everything, including a sort of shorthand notation to save time and space. The notation that comes with particular properties cuts down on misinterpretation because it’s very specific and universally known. (I give the guidelines for doing operations like addition, subtraction, multiplication, and division in Chapter 1.) In this chapter, you see the specific rules that apply when you use grouping symbols and rearrange terms.
Getting a Grip on Grouping Symbols
The most commonly used grouping symbols in algebra are (in order from most to least common):
Parentheses ( )
Brackets [ ]
Braces { }
Fraction lines /
Radicals
Absolute value symbols | |
Here’s what you need to know about grouping symbols: You must compute whatever is inside them (or under or over, in the case of the fraction line) first, before you can use that result to solve the rest of the problem. If what’s inside isn’t or can’t be simplified into one term, then anything outside the grouping symbol that multiplies one of the terms has to multiply them all — that’s the distributive property, which I cover in the very next section.
Q. 16 – (4 + 2) =
A. 10. Add the 4 and 2; then subtract the result from the 16: 16 – (4 + 2) = 16 – 6 = 10
Q. Simplify 2[6 – (3 – 7)].
A. 20. First subtract the 7 from the 3; then subtract the –4 from the 6 by changing it to an addition problem. You can then multiply the 2 by the 10: 2[6 – (3 – 7)] = 2[6 – (–4)] = 2[6 + 4] = 2[10] = 20
Q.
A. 8. Combine what’s in the absolute value and parentheses first, before combining the results:
When you get to the three terms with subtract and add, 1 – 11 + 18, you always perform the operations in order, reading from left to right. See Chapter 6 for more on this process, called the order of operations.
Q.
A. 2. You have to complete the work in the denominator first before dividing the 32 by that result:
1. 3(2 – 5) + 14 =
Solve It
2. 4[3(6 – 8) + 2(5 + 9)] – 11 =
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3. 5{8[2 + (6 – 3)] – 4} =
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6.
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Distributing the Wealth
The distributive property is used to perform an operation on each of the terms within a grouping symbol. The following rules show distributing multiplication over addition and distributing multiplication over subtraction:
a(b + c) = a × b + a × c and a(b – c) = a × b – a × c
Q. 3(6 – 4) =
A. 6. First, distribute the 3 over the 6 – 4: 3(6 – 4) = 3 × 6 – 3 × 4 = 18 – 12 = 6. Another (simpler) way to get the correct answer is just to subtract the 4 from the 6 and then multiply: 3(2) = 6. However, when you can’t or don’t want to combine what’s in the grouping symbols, you use the distributive property.
Q.
A. 5a – 1
7. 4(7 + y) =
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8. –3(x – 11) =
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11.
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12.
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Making Associations Work
The associative rule in math says that in addition and multiplication problems, you can change the association, or groupings, of three or more numbers and not change the final result. The associative rule looks like the following:
a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c
This rule is special to addition and multiplication. It doesn’t work for subtraction or division. You’re probably wondering why even use this rule? Because it can sometimes make the computation easier.
Instead of doing 5 + (–5 + 17), change it to [5 + (–5)] + 17 = 0 + 17 = 17.
Instead of , do .
Q. –14 + (14 + 23) =
A. 23. Reassociate the terms and then add the first two together: –14 + (14 + 23) = (–14 + 14) + 23 = 0 + 23 = 23.
Q. 4(5 × 6) =
A. 120. You can either multiply the way the problem is written, 4(5 × 6) = 4(30) = 120, or you can reassociate and multiply the first two factors first: (4 × 5) 6 = (20)6 = 120.
13. 16 + (–16 + 47) =
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14. (5 – 13) + 13 =
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15.
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16.
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Computing by Commuting
The commutative property of addition and multiplication says that the order that you add or multiply numbers doesn’t matter. Be careful, though, because the order of subtraction and division does matter. You get the same answer whether you 3 × 4 or 4 × 3. The rule looks like the following:
a + b = b + a and a × b = b × a
You can use this rule to your advantage when doing math computations. In the following two examples, the associative rule finishes off the problems after changing the order.
Q.
A. . You don’t really want to multiply fractions unless necessary. Notice that the first and last factors are multiplicative inverses of one another: . The second and last factors were reversed.
Q. –3 + 16 + 303 =
A. 316. The second and last terms are reversed, and then the first two terms are grouped.
–3 + 16 + 303 = –3 + 303 + 16 = (–3 + 303) + 16 = 300 + 16 = 316.
17. 8 + 5 + (–8) =
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18. 5 × 47 × 2 =
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19.
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20. –23 + 47 + 23 – 47 + 8 =
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Answers to Problems on Algebraic Properties
This section provides the answers (in bold) to the practice problems in this chapter.
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Chapter 3
Making Fractions and Decimals Behave
In This Chapter
Simplifying and changing fractions
Making proportions work for you
Operating on fractions
You can try to run and hide, but you may as well face it. Fractions are here to stay. People don’t usually eat a whole pizza, buy furniture that’s exactly 5 feet long, or grow to be an even number of inches tall. Fractions are not only useful, but they’re also an essential part of everyday life.
Fractions are equally as important in algebra. Many times, to complete a problem, you have to switch from one form of a fraction to another. This chapter provides you plenty of opportunities to work out all your fractional frustration.