Download Chapter 2 Section 5

Chapter 2
Equations,
Inequalities and
Problem Solving
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Bellwork:
1. If the perimeter of the
following pentagon is
28cm, find the length of
each side.
2. The perimeter of the
following triangle is 35m.
Find the length of each side.
x cm
x cm
4 cm
2x cm
8 cm 2x cm
xm
(2x+1) m
13 m
6 m 16 m
(3x-2) m
x cm
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2
2.5
An Introduction to
Problem Solving
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Objectives:
Translate
a problem to an
equation, then use the
equation to solve the problem
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Writing Phrases as Algebraic Expressions
Addition Subtraction
(+)
(-)
sum
difference
minus
plus
added to
subtracted
from
Multiplication Division
(∙)
(÷)
product
quotient
times
divide
multiply
into
more than
increased
by
twice
ratio
less than
decreased by of
total
less
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Equal Sign
equals
gives
is/was/
should be
yields
divided by amounts to
represents
is the same
as
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Strategy for Problem Solving
1. UNDERSTAND the problem.
• Read and reread the problem.
• Choose a variable to represent the
unknown.
• Construct a drawing.
2. TRANSLATE the problem into an equation.
3. SOLVE the equation.
4. INTERPRET the result:
• Check proposed solution in problem.
• State your conclusion.
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Example 1
Twice a number plus 3 is the same as the number minus 6.
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Example 1
Twice a number plus 3 is the same as the number minus 6.
2x
3 
x 6
2x  3  x  6
2x  3  x  x  6  x
x  3  6
x  3  3  6  3
x  9
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Finding an Unknown Number
Example
The product of twice a number and three is the same as the
difference of five times the number and ¾. Find the number.
1. Understand
Read and reread the problem. If we let
x = the unknown number, then “twice a number” translates to 2x,
“the product of twice a number and three” translates to 2x · 3,
“five times the number” translates to 5x, and
“the difference of five times the number and ¾” translates to 5x – ¾.
continued
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continued
2. Translate
The product of
is the same as
twice a
number
2x
the difference of
and 3
·
3
=
5 times the
number
5x
and ¾
–
¾
continued
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continued
3. Solve
2x · 3 = 5x – ¾
6x = 5x – ¾
6x + (–5x) = 5x + (–5x) – ¾
x = –¾
4. Interpret
Check: Replace “number” in the original statement of the problem
with –¾. The product of twice –¾ and 3 is 2(–¾)(3) = –4.5. The
difference of five times –¾ and ¾ is 5(–¾) –¾ = – 4.5. We get the
same results for both portions.
State: The number is –¾.
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Example 2
A car rental agency advertised renting a Buick Century for
$24.95 per day and $0.29 per mile. If you rent this car for 2
days, how many whole miles can you drive on a $100
budget?
x = the number of whole miles driven, then
0.29x = the cost for mileage driven
2(24.95) + 0.29x = 100
continued
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continued
2(24.95) + 0.29x = 100
49.90 + 0.29x = 100
49.90 – 49.90 + 0.29x = 100 – 49.90
0.29x = 50.10
0.29 x 50.10

0.29
0.29
x  172.75
continued
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continued
Check: Recall that the original statement of the
problem asked for a “whole number” of miles. If we
replace “number of miles” in the problem with 173,
then 49.90 + 0.29(173) = 100.07, which is over our
budget. However, 49.90 + 0.29(172) = 99.78, which is
within the budget.
State: The maximum number of whole
number miles is 172.
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Example 3
The measure of the second angle of a triangle is twice the
measure of the smallest angle. The measure of the third
angle of the triangle is three times the measure of the
smallest angle. Find the measures of the angles.
x°
Draw a diagram.
Let x = degree measure of smallest angle
2x = degree measure of second angle
3x = degree measure of third angle
2x°
3x°
continued
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continued
Recall that the sum of the measures of the angles of a
triangle equals 180.
measure of
first angle
x
measure of
second
angle
+
2x +
measure of
third angle
equals
180
3x
=
180
continued
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continued
x + 2x + 3x = 180
6x = 180
6 x 180

6
6
x = 30
Check: If x = 30, then 2x = 2(30) = 60 and 3x = 3(30) = 90
The sum of the angles is 30 + 60 + 90 = 180.
State:
The smallest angle is 30º, the second
angle is 60º, and the third angle is 90º.
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Example 4
The sum of three consecutive even integers is 252. Find
the integers.
x = the first even integer
x + 2 = next even integer
x + 4 = next even integer
Translate:
x + x + 2 + x + 4 = 252
continued
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continued
The sum of three consecutive even integers is 252. Find
the integers.
x + x + 2 + x + 4 = 252
3x + 6 = 252
3x = 246
3x 246

3
3
x  82
The integers are 82, 84 and 86.
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Closure:
What are the steps to working
with a word problem?
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