Download mgalg2_02_02

Copyright
Copyright
© 2011
© 2011
Pearson
Pearson
Education,
Education,
Inc. Publishing
Inc. Publishing
as Prentice
as Prentice
Hall Hall
Chapter 2
Equations, Inequalities,
and Problem Solving
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall
2.2
An Introduction to
Problem Solving
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall
Consecutive, even, and odd integers and their
representations.
+2
+1
Consecutive Integers:
x
x+1
x+2
+4
Consecutive Even Integers:
+2
x
x+2
Consecutive Odd Integers:
+4
+2
x
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall
x+2
x+4
x+4
Example 1
Write the following as an algebraic expression. Then
simplify.
The sum of three consecutive odd integers, if x is the
first consecutive integer.
Solution
first
plus next odd plus next odd
In words:
integer
integer
integer
Translate:
x
Then
x + (x + 2) + (x + 4) = x + x + 2 + x + 4
= 3x + 6
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall
+
(x + 2) +
(x + 4)
General Strategy for Problem Solving
1) UNDERSTAND the problem. During this step, become
comfortable with the problem. Some way of doing this are:
• Read and reread the problem.
• Propose a solution and check.
• Construct a drawing.
• Choose a variable to represent the unknown.
2) TRANSLATE the problem into an equation.
3) SOLVE the equation.
4) INTERPRET the result. Check the proposed solution in the
stated problem and state your conclusion.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall
Example 3
Find three numbers such that the second is 3 more than
twice the first number, and the third number is four
times the first number. The sum of the three numbers is
164.
Solution
UNDERSTAND the problem. Read and reread the
problem. If we let
x = the first number, then
2x + 3 = the second number
4x = the third number
continued
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall
2. TRANSLATE
In words: first
number
added
to
second
number
added
to
third
number
is
164
Translate: x
+
(2x + 3) +
4x
= 164
3. Solve the equation.
x + (2x + 3) + 4x = 164
x + 2x + 4x + 3 = 164
Remove parentheses.
7x + 3 = 164
Combine like terms.
7x = 161
Subtract 3 from both sides.
x = 23
Divide both sides by 7.
4. INTERPRET. Here, we check our work and state the
solution;
x = 23, the second number 2x + 3 = 2(23) + 3 = 49,
and the third number 4x = 4(23) = 92.
continued
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall
Check: Is the second number 3 more than twice the first
number? Yes, since 3 more than twice 23 is 46 + 3, or
49. Also, their sum, 23 + 49 + 92 = 164, is the
required sum.
State: The three numbers are 23, 49, and 92.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall
Example 6
Kelsey Ohleger was helping her friend Benji Burnstine
study for an algebra exam. Kelsey told Benji that her
last three latest art history quiz scores are three
consecutive even integers whose sum is 264. Help Benji
find the scores.
Solution
UNDERSTAND the problem. Read and reread the
problem. We are looking for consecutive integers, let
x = the first integer. Then
x + 2 = the second consecutive even integer
x + 4 = the third consecutive even integer.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall
continued
2. TRANSLATE
first
In words: integer
+
second even
integer
+
third even
integer
=
264
Translate: x
+
(x + 2)
+ (x + 4) = 264
3. SOLVE.
x + (x + 2) + (x + 4) = 264
3x + 6 = 264
Combine like terms.
3x = 258
Subtract 6 from both sides.
x = 86
Divide both sides by 3.
4. INTERPRET. If x = 86, then x + 2 = 86 + 2 or 88, and
x + 4 = 86 + 4 or 90.
continued
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall
Check: The numbers 86, 88, and 90 are three
consecutive even integers. Their sum is 264, the
required sum.
State: Kelsey’s art history quiz scores are 86, 88 and
90.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall