3: More on Solving Equations and Problem Solving
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104
Chapter 3 • Equations, Inequalities, and Problem Solving
Use the addition or subtraction property of equality to isolate a term that contains
the variable on one side and a constant on the other side of the equal sign.
Step 3 Use the multiplication or division property of equality to make the coefficient of the
variable one.
Step 2
The next examples illustrate this step-by-step process for solving equations. Study these
examples carefully and be sure that you understand each step taken in the solution process.
Classroom Example
Solve 7m ϩ 1 Ϫ 3m ϭ 21.
Solve 5y Ϫ 4 ϩ 3y ϭ 12 .
EXAMPLE 1
Solution
5y Ϫ 4 ϩ 3y ϭ 12
8y Ϫ 4 ϭ 12
8y Ϫ 4 ϩ 4 ϭ 12 ϩ 4
8y ϭ 16
8y
16
ϭ
8
8
Combine similar terms on the left side
Add 4 to both sides
Divide both sides by 8
yϭ2
The solution set is 526. You can do the check alone now!
Classroom Example
Solve 8w ϩ 5 ϭ 2w Ϫ 3.
Solve 7x Ϫ 2 ϭ 3x ϩ 9.
EXAMPLE 2
Solution
Notice that both sides of the equation are in simplified form; thus we can begin by applying
the subtraction property of equality.
7x Ϫ 2 ϭ 3x ϩ 9
7x Ϫ 2 Ϫ 3x ϭ 3x ϩ 9 Ϫ 3x
Subtract 3x from both sides
4x Ϫ 2 ϭ 9
4x Ϫ 2 ϩ 2 ϭ 9 ϩ 2
Add 2 to both sides
4x ϭ 11
4x
11
ϭ
4
4
xϭ
The solution set is e
Classroom Example
Solve 3d ϩ 1 ϭ 4d Ϫ 3.
EXAMPLE 3
Divide both sides by 4
11
4
11
f.
4
Solve 5n ϩ 12 ϭ 9n Ϫ 16.
Solution
5n ϩ 12 ϭ 9n Ϫ 16
5n ϩ 12 Ϫ 9n ϭ 9n Ϫ 16 Ϫ 9n
Subtract 9n from both sides
Ϫ4n ϩ 12 ϭ Ϫ16
Ϫ4n ϩ 12 Ϫ 12 ϭ Ϫ16 Ϫ 12
Subtract 12 from both sides
3.3 • More on Solving Equations and Problem Solving
105
Ϫ4n ϭ Ϫ28
Ϫ4n
Ϫ28
ϭ
Ϫ4
Ϫ4
Divide both sides by Ϫ4
nϭ7
The solution set is 576.
Classroom Example
Solve 2m ϩ 5 ϭ 2m Ϫ 3.
EXAMPLE 4
Solve 3x ϩ 8 ϭ 3x Ϫ 2.
Solution
3x ϩ 8 ϭ 3x Ϫ 2
3x ϩ 8 Ϫ 3x ϭ 3x Ϫ 2 Ϫ 3x
8 ϭ Ϫ2
Subtract 3x from both sides
False statement
Since we obtained an equivalent equation that is a false statement, there is no value of x that
will make the equation a true statement. When the equation is not true under any condition,
then the equation is called a contradiction. The solution set for an equation that is a contradiction is the empty or null set, and it is symbolized by л.
Classroom Example
Solve 3n ϩ 4 ϭ 8n ϩ 4 Ϫ 5n.
EXAMPLE 5
Solve 4x ϩ 6 Ϫ x ϭ 3x ϩ 6.
Solution
4x ϩ 6 Ϫ x ϭ 3x ϩ 6
3x ϩ 6 ϭ 3x ϩ 6
Combine similar terms on the left side
3x ϩ 6 Ϫ 3x ϭ 3x ϩ 6 Ϫ 3x
Subtract 3x from both sides
6ϭ6
True statement
Since we obtained an equivalent equation that is a true statement, any value of x will make
the equation a true statement. When an equation is true for any value of the variable, the equation is called an identity. The solution set for an equation that is an identity is the set of all
real numbers. We will denote the set of all real numbers as {all reals}.
Word Problems
As we expand our skills for solving equations, we also expand our capabilities for solving
word problems. No one definite procedure will ensure success at solving word problems, but
the following suggestions can be helpful.
Suggestions for Solving Word Problems
1. Read the problem carefully, and make sure that you understand the meanings of all
the words. Be especially alert for any technical terms in the statement of the
problem.
2. Read the problem a second time (perhaps even a third time) to get an overview of
the situation being described and to determine the known facts as well as what is to
be found.
3. Sketch any figure, diagram, or chart that might be helpful in analyzing the problem.
(continued)
106
Chapter 3 • Equations, Inequalities, and Problem Solving
4. Choose a meaningful variable to represent an unknown quantity in the problem
(perhaps t if time is an unknown quantity); represent any other unknowns in terms
of that variable.
5. Look for a guideline that you can use to set up an equation. A guideline might be a
formula such as distance equals rate times time or a statement of a relationship such
as the sum of the two numbers is 28. A guideline may also be indicated by a figure
or diagram that you sketch for a particular problem.
6. Form an equation that contains the variable and that translates the conditions of the
guideline from English into algebra.
7. Solve the equation and use the solution to determine all facts requested in the
problem.
8. Check all answers by going back to the original statement of the problem and
verifying that the answers make sense.
If you decide not to check an answer, at least use the reasonableness-of-answer idea as a partial check. That is to say, ask yourself the question: Is this answer reasonable? For example,
if the problem involves two investments that total $10,000, then an answer of $12,000 for one
investment is certainly not reasonable.
Now let’s consider some examples and use these suggestions as you work them out.
Consecutive Number Problems
Some problems involve consecutive numbers or consecutive even or odd numbers. For
instance, 7, 8, 9, and 10 are consecutive numbers. To solve these applications, you must know
how to represent consecutive numbers with variables. Let n represent the first number. For
consecutive numbers, the next number is 1 more and is represented by n ϩ 1. To continue,
we add 1 to each preceding expression, obtaining the representations shown here:
7
8
9
10
n
nϩ1
nϩ2
nϩ3
The pattern is somewhat different for consecutive even or odd numbers. For example, 2, 4, 6,
and 8 are consecutive even numbers. Let n represent the first even number; then n ϩ 2 represents the next even number. To continue, we add 2 to each preceding expression, obtaining
these representations:
2
4
6
8
n
nϩ2
nϩ4
nϩ6
Consecutive odd numbers have the same pattern of adding 2 to each preceding expression
because consecutive odd numbers are two odd numbers with one and only one whole number between them.
Classroom Example
Find two consecutive even numbers
whose sum is 42.
EXAMPLE 6
Find two consecutive even numbers whose sum is 74.
Solution
Let n represent the first number; then n ϩ 2 represents the next even number. Since their sum
is 74, we can set up and solve the following equation:
n ϩ 1n ϩ 22 ϭ 74
2n ϩ 2 ϭ 74
3.3 • More on Solving Equations and Problem Solving
107
2n ϩ 2 Ϫ 2 ϭ 74 Ϫ 2
2n ϭ 72
2n 72
ϭ
2
2
n ϭ 36
If n ϭ 36, then n ϩ 2 ϭ 38; thus, the numbers are 36 and 38.
✔ Check
To check your answers for Example 6, determine whether the numbers satisfy the conditions
stated in the original problem. Because 36 and 38 are two consecutive even numbers, and
36 ϩ 38 ϭ 74 (their sum is 74), we know that the answers are correct.
The fifth entry in our list of problem-solving suggestions is to look for a guideline that
can be used to set up an equation. The guideline may not be stated explicitly in the problem
but may be implied by the nature of the problem. Consider the following example.
Classroom Example
Tyron sells appliances on a salaryplus-commission basis. He receives
a monthly salary of $550 and a
commission of $95 for each appliance
that he sells. How many appliances
must he sell in a month to earn a total
monthly salary of $1,120.
EXAMPLE 7
Barry sells bicycles on a salary-plus-commission basis. He receives a weekly salary of $300
and a commission of $15 for each bicycle that he sells. How many bicycles must he sell in a week
to earn a total weekly salary of $750?
Solution
Let b represent the number of bicycles to be sold in a week. Then 15b represents his commission for those bicycles. The guideline “fixed salary plus commission equals total weekly
salary” generates the following equation:
Fixed salary ϩ Commission ϭ Total weekly salary
$300
ϩ
15b
ϭ
$750
Solving this equation yields
300 ϩ 15b Ϫ 300 ϭ 750 Ϫ 300
15b ϭ 450
15b 450
ϭ
15
15
b ϭ 30
Barry must sell 30 bicycles per week. (Does this number check?)
Geometric Problems
Sometimes the guideline for setting up an equation to solve a problem is based on a geometric relationship. Several basic geometric relationships pertain to angle measure. Let’s state
some of these relationships and then consider some examples.
1. Two angles for which the sum of their measure is 90Њ (the symbol Њ indicates degrees)
are called complementary angles.
2. Two angles for which the sum of their measure is 180Њ are called supplementary
angles.
3. The sum of the measures of the three angles of a triangle is 180Њ.
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Chapter 3 • Equations, Inequalities, and Problem Solving
Classroom Example
One of two supplementary angles is
26° smaller than the other. Find the
measure of each of the angles.
EXAMPLE 8
One of two complementary angles is 14Њ larger than the other. Find the measure of each of
the angles.
Solution
If we let a represent the measure of the smaller angle, then a ϩ 14 represents the measure of
the larger angle. Since they are complementary angles, their sum is 90°, and we can proceed
as follows:
a ϩ 1a ϩ 142 ϭ 90
2a ϩ 14 ϭ 90
2a ϩ 14 Ϫ 14 ϭ 90 Ϫ 14
2a ϭ 76
2a 76
ϭ
2
2
a ϭ 38
If a ϭ 38, then a ϩ 14 ϭ 52, and the angles measure 38Њ and 52Њ.
Classroom Example
Find the measures of the three angles
of a triangle if the second angle is
twice the first angle, and the third
angle is half the second angle.
EXAMPLE 9
Find the measures of the three angles of a triangle if the second is three times the first and
the third is twice the second.
Solution
If we let a represent the measure of the smallest angle, then 3a and 2(3a) represent the measures of the other two angles. Therefore, we can set up and solve the following equation:
a ϩ 3a ϩ 213a2 ϭ 180
a ϩ 3a ϩ 6a ϭ 180
10a ϭ 180
10a 180
ϭ
10
10
a ϭ 18
If a ϭ 18, then 3a ϭ 54 and 213a2 ϭ 108. So the angles have measures of 18Њ, 54Њ, and 108Њ.
Concept Quiz 3.3
For Problems 1–10, answer true or false.
1. If n represents a whole number, then n ϩ 1 would represent the next consecutive whole
number.
2. If n represents an odd whole number, then n ϩ 1 would represent the next consecutive
odd whole number.
3. If n represents an even whole number, then n ϩ 2 would represent the next consecutive
even whole number.
4. The sum of the measures of two complementary angles is 90Њ.
5. The sum of the measures of two supplementary angles is 360Њ.
6. The sum of the measures of the three angles of a triangle is 120Њ.
7. When checking word problems, it is sufficient to check the solution in the equation.
8. For a word problem, the reasonableness of an answer is appropriate as a partial check.
9. For a conditional equation, the solution set is the set of all real numbers.
10. The solution set for an equation that is a contradiction is the null set.
3.3 • More on Solving Equations and Problem Solving
109
Problem Set 3.3
For Problems 1– 32, solve each equation. (Objectives 1–3)
1. 2x ϩ 7 ϩ 3x ϭ 32
2. 3x ϩ 9 ϩ 4x ϭ 30
30. 4x Ϫ 3 ϩ 2x ϭ 8x Ϫ 3 Ϫ x
31. Ϫ7 Ϫ 2n Ϫ 6n ϭ 7n Ϫ 5n ϩ 12
32. Ϫ3n ϩ 6 ϩ 5n ϭ 7n Ϫ 8n Ϫ 9
3. 7x Ϫ 4 Ϫ 3x ϭ Ϫ36
4. 8x Ϫ 3 Ϫ 2x ϭ Ϫ45
5. 3y Ϫ 1 ϩ 2y Ϫ 3 ϭ 4
6. y ϩ 3 ϩ 2y Ϫ 4 ϭ 6
Solve each word problem by setting up and solving an algebraic equation. (Objectives 4 and 5)
33. The sum of a number plus four times the number is 85.
What is the number?
7. 5n Ϫ 2 Ϫ 8n ϭ 31
34. A number subtracted from three times the number
yields 68. Find the number.
8. 6n Ϫ 1 Ϫ 10n ϭ 51
35. Find two consecutive odd numbers whose sum is 72.
9. Ϫ2n ϩ 1 Ϫ 3n ϩ n Ϫ 4 ϭ 7
10. Ϫn ϩ 7 Ϫ 2n ϩ 5n Ϫ 3 ϭ Ϫ6
11. 3x ϩ 4 ϭ 2x Ϫ 5
36. Find two consecutive even numbers whose sum is 94.
37. Find three consecutive even numbers whose sum is 114.
12. 5x Ϫ 2 ϭ 4x ϩ 6
13. 5x Ϫ 7 ϭ 6x Ϫ 9
14. 7x Ϫ 3 ϭ 8x Ϫ 13
15. 6x ϩ 1 ϭ 3x Ϫ 8
16. 4x Ϫ 10 ϭ x ϩ 17
17. 7y Ϫ 3 ϭ 5y ϩ 10
18. 8y ϩ 4 ϭ 5y Ϫ 4
19. 8n Ϫ 2 ϭ 8n Ϫ 7
20. 7n Ϫ 10 ϭ 9n Ϫ 13
21. Ϫ2x Ϫ 7 ϭ Ϫ3x ϩ 10
38. Find three consecutive odd numbers whose sum is 159.
39. Two more than three times a certain number is the same
as 4 less than seven times the number. Find the number.
40. One more than five times a certain number is equal to
eleven less than nine times the number. What is the
number?
41. The sum of a number and five times the number equals
eighteen less than three times the number. Find the
number.
42. One of two supplementary angles is five times as large
as the other. Find the measure of each angle.
22. Ϫ4x ϩ 6 ϭ Ϫ5x Ϫ 9
23. Ϫ3x ϩ 5 ϭ Ϫ5x Ϫ 8
24. Ϫ4x ϩ 7 ϭ Ϫ4x ϩ 4
25. Ϫ7 Ϫ 6x ϭ 9 Ϫ 9x
26. Ϫ10 Ϫ 7x ϭ 14 Ϫ 12x
27. 2x Ϫ 1 Ϫ x ϭ x Ϫ 1
28. 3x Ϫ 4 Ϫ 4x ϭ Ϫ5x ϩ 4x Ϫ 4
29. 5n Ϫ 4 Ϫ n ϭ Ϫ3n Ϫ 6 ϩ n
43. One of two complementary angles is 6Њ smaller than
twice the other angle. Find the measure of each angle.
44. If two angles are complementary and the difference
between their measures is 62Њ, find the measure of each
angle.
45. If two angles are supplementary and the larger angle is
20Њ less than three times the smaller angle, find the measure of each angle.
46. Find the measures of the three angles of a triangle if the
largest is 14Њ less than three times the smallest, and the
other angle is 4Њ larger than the smallest.
110
Chapter 3 • Equations, Inequalities, and Problem Solving
47. One of the angles of a triangle has a measure of 40Њ.
Find the measures of the other two angles if the difference between their measures is 10Њ.
The other side is 10 yards longer than the shortest side.
Find the lengths of the three sides of the lot.
48. Jesstan worked as a telemarketer on a salary-pluscommission basis. He was paid a salary of $300 a week
and a $12 commission for each sale. If his earnings for
the week were $960, how many sales did he make?
49. Marci sold an antique vase in an online auction for
$69.00. This was $15 less than twice the amount she
paid for it. What price did she pay for the vase?
50. A set of wheels sold in an online auction for $560. This
was $35 more than three times the opening bid. How
much was the opening bid?
51. Suppose that Bob is paid two times his normal hourly
rate for each hour he works over 40 hours in a week.
Last week he earned $504 for 48 hours of work. What
is his hourly wage?
52. Last week on an algebra test, the highest grade was 9
points less than three times the lowest grade. The sum
of the two grades was 135. Find the lowest and highest
grades on the test.
53. At a university-sponsored concert, there were three
times as many women as men. A total of 600 people
attended the concert. How many men and how many
women attended?
54. Suppose that a triangular lot is enclosed with 135 yards
of fencing (see Figure 3.1). The longest side of the lot is
5 yards longer than twice the length of the shortest side.
Figure 3.1
55. The textbook for a biology class cost $15 more than
twice the cost of a used textbook for college algebra. If
the cost of the two books together is $129, find the cost
of the biology book.
56. A nutrition plan counts grams of fat, carbohydrates, and
fiber. The grams of carbohydrates are to be 15 more than
twice the grams of fat. The grams of fiber are to be three
less than the grams of fat. If the grams of carbohydrate,
fat, and fiber must total 48 grams for a dinner meal, how
many grams of each would be in the meal?
57. At a local restaurant, $275 in tips is to be shared between
the server, bartender, and busboy. The server gets $25
more than three times the amount the busboy receives.
The bartender gets $50 more than the amount the busboy
receives. How much will the server receive?
Additional word problems can be found in Appendix B.
All the problems in the Appendix marked as (3.3) are
appropriate for this section.
Thoughts Into Words
58. Give a step-by-step description of how you would solve
the equation 3x ϩ 4 ϭ 5x Ϫ 2.
59. Suppose your friend solved the problem find two consecutive odd integers whose sum is 28 as follows:
x ϩ 1x ϩ 12 ϭ 28
2x ϭ 27
27
1
ϭ 13
2
2
1
She claims that 13 will check in the equation. Where
2
has she gone wrong, and how would you help her?
xϭ
Further Investigations
60. Solve each of these equations.
(a) 7x Ϫ 3 ϭ 4x Ϫ 3
(b) Ϫx Ϫ 4 ϩ 3x ϭ 2x Ϫ 7
(c) Ϫ3x ϩ 9 Ϫ 2x ϭ Ϫ5x ϩ 9
(d) 5x Ϫ 3 ϭ 6x Ϫ 7 Ϫ x
(e) 7x ϩ 4 ϭ Ϫx ϩ 4 ϩ 8x
(f) 3x Ϫ 2 Ϫ 5x ϭ 7x Ϫ 2 Ϫ 5x
(g) Ϫ6x Ϫ 8 ϭ 6x ϩ 4
(h) Ϫ8x ϩ 9 ϭ Ϫ8x ϩ 5
61. Make up an equation whose solution set is the null set and
explain why.
62. Make up an equation whose solution set is the set of all real
numbers and explain why.
3.4 • Equations Involving Parentheses and Fractional Forms
Answers to the Concept Quiz
1. True
2. False
3. True
9. False
10. True
3.4
4. True
5. False
6. False
7. False
111
8. True
Equations Involving Parentheses and Fractional Forms
OBJECTIVES
1
Solve ﬁrst-degree equations that involve the use of the distributive
property
2
Solve ﬁrst-degree equations that involve fractional forms
3
Solve a variety of word problems involving ﬁrst-degree equations
We will use the distributive property frequently in this section as we add to our techniques
for solving equations. Recall that in symbolic form the distributive property states that
a(b ϩ c) ϭ ab ϩ ac. Consider the following examples, which illustrate the use of this property to remove parentheses. Pay special attention to the last two examples, which involve a
negative number in front of the parentheses.
иxϩ3и2
5 и y Ϫ 5 и 3
3(x ϩ 2) ϭ
3
5(y Ϫ 3) ϭ
2(4x ϩ 7) ϭ
ϭ 3x ϩ 6
ϭ 5y Ϫ 15
a(b Ϫ c) ϭ ab Ϫ ac
2(4x) ϩ 2(7) ϭ 8x ϩ 14
Ϫ1(n ϩ 4) ϭ (Ϫ1)(n) ϩ (Ϫ1)(4) ϭ Ϫn Ϫ 4
Ϫ6(x Ϫ 2) ϭ (Ϫ6)(x) Ϫ (Ϫ6)(2) ϭ Ϫ6x ϩ 12
Do this step mentally!
It is often necessary to solve equations in which the variable is part of an expression enclosed
in parentheses. We use the distributive property to remove the parentheses, and then we proceed in the usual way. Consider the next examples. (Notice that we are beginning to show
only the major steps when solving an equation.)
Classroom Example
Solve 2(d Ϫ 5) ϭ 5.
Solve 31x ϩ 22 ϭ 23.
EXAMPLE 1
Solution
31x ϩ 22 ϭ 23
3x ϩ 6 ϭ 23
Applied distributive property to left side
3x ϭ 17
xϭ
Subtracted 6 from both sides
17
3
The solution set is e
Divided both sides by 3
17
f.
3
112
Chapter 3 • Equations, Inequalities, and Problem Solving
Classroom Example
Solve 3(m Ϫ 1) ϭ 6 (m ϩ 2).
EXAMPLE 2
Solve 41x ϩ 32 ϭ 21x Ϫ 62 .
Solution
41x ϩ 32 ϭ 21x Ϫ 62
4x ϩ 12 ϭ 2x Ϫ 12
2x ϩ 12 ϭ Ϫ12
2x ϭ Ϫ24
x ϭ Ϫ12
Applied distributive property on each side
Subtracted 2x from both sides
Subtracted 12 from both sides
Divided both sides by 2
The solution set is 5Ϫ126.
It may be necessary to remove more than one set of parentheses and then to use the
distributive property again to combine similar terms. Consider the following two examples.
Classroom Example
Solve 2(t ϩ 4) ϩ 6(t ϩ 5) ϭ 32.
EXAMPLE 3
Solve 51w ϩ 32 ϩ 31w ϩ 12 ϭ 14.
Solution
51w ϩ 32 ϩ 31w ϩ 12 ϭ 14
5w ϩ 15 ϩ 3w ϩ 3 ϭ 14
8w ϩ 18 ϭ 14
8w ϭ Ϫ4
4
8
1
wϭϪ
2
wϭϪ
Applied distributive property
Combined similar terms
Subtracted 18 from both sides
Divided both sides by 8
Reduced
1
The solution set is eϪ f .
2
Classroom Example
Solve 4(w Ϫ 3) Ϫ 8(w Ϫ 2) ϭ 9.
EXAMPLE 4
Solve 61x Ϫ 72 Ϫ 21x Ϫ 42 ϭ 13.
Solution
61x Ϫ 72 Ϫ 21x Ϫ 42 ϭ 13
6x Ϫ 42 Ϫ 2x ϩ 8 ϭ 13
4x Ϫ 34 ϭ 13
4x ϭ 47
47
xϭ
4
The solution set is e
47
f.
4
Be careful with this sign!
Distributive property
Combined similar terms
Added 34 to both sides
Divided both sides by 4