Download Chapter 9.4

Chapter 9.4
More Problem Solving
1
Words to Symbols
Addition
Subtraction
Multiplication Division
Equals
Sum
Difference of
Product
Quotient
Gives
Plus
Minus
Times
Divide
Is/Was/Should be
Added To
Subtracted From
Multiply
Into
Yields
More Than
Less Than
Twice
Ratio
Amounts to
Increased by
Decreased By
Of
Divided By
Represents
Total
Less
Is the Same As
Solving Word Problems
• Read the problem
– Reread
– Find the question to be answered
– Draw, if possible
– Identify quantities
– Assign variables to unknown quantities
• Translate the problem to an equation
• Solve the equation
• Look at the results – do they make sense?
Practice
• Twice a number added to seven is the same as three
subtracted from the number
Solution
• 2x + 7 = x – 3
• x = -10
Hint
• When checking your answer, go back to the original problem,
if possible, not just your equation
Example
• Twice the sum of a number and 4 is the same as four times
the number decreased by 12
Solution
• 2 (x + 4) = 4x – 12
x = 10
Example
• Three times the difference of a number and 5 is the same as
twice the number decreased by 3.
Solution
• 3 (x-5) = 2x – 3
3x – 15 = 2x – 3
x = 12
Example
• A ten foot board is to be cut into two pieces so that the length
of the longer piece is 4 times the length of the shortest. Find
the length of each piece
Solution
• x is shorter piece
• 4x is longer piece
• x + 4x = 10
x=2
Example
• A computer science major works part time fixing computers.
He charges $20 to come to your home and then $25 per hour.
During one month he visited 10 homes and his income was
$575. How many hours did he spend fixing computers?
Solution
• A computer science major works part time fixing computers.
He charges $20 to come to your home and then $25 per hour.
During one month he visited 10 homes and his income was
$575. How many hours did he spend fixing computers?
• For each house he gets $20, without hours.
• He went to 10 homes, so $200
• He earned $575; $200 of that was for visiting, so 375 was for
hourly work. Since he earns $25 per hour, 375 = 25H, where
H is the number of hours
= 375/15 = 35 hours
Example
• The two walls of the Vietnam Memorial in DC form an
isosceles triangle (two sides and two angles the same).
The measure of the third angle is 97.5 degrees more than
either of the two equal angles. Find the third angle.
Solution
• The two walls of the Vietnam Memorial in DC form an
isosceles triangle (two sides and two angles the same).
The measure of the third angle is 97.5 degrees more than
either of the two equal angles. Find the third angle.
180 degrees in a triangle
The third angle is 97.5 deg more than either of the other two,
so the third angle is 97.5 + a, where a is the other two
We have 180 = 97.5 + a + 2a = 97.5 + 3a
3a = 180 – 97.5 = 82.5
a = 27.5
Consecutive Integers
• Two states have area codes that are consecutive odd
integers. If the sum of these integers is 1208, what are the
integers?
Solution
• Two states have area codes that are consecutive odd
integers. If the sum of these integers is 1208, what are the
integers?
Consecutive integers are n and n + 2
n + n + 2 = 1208
2n = 1206
n = 603, n + 2 = 605
Example
Two consecutive odd numbered classrooms have the sum of
their numbers equal 804. Find the two class numbers
Solution
Two consecutive odd numbered classrooms have the sum of
their numbers equal 804. Find the two class numbers
An odd number is written as 2n + 1. The next consecutive odd is
2(n+1) + 1 = 2n + 3
2n + 1 + 2n + 3 = 4n + 4 = 804
n = 800/4 = 200
Number = 2(200)+1 = 401 and 403
Example
• The code to unlock your gym locker is three consecutive even
numbers whose sum is 240. Find the three numbers.
Solution
• The code to unlock your gym locker is three consecutive even
numbers whose sum is 240. Find the three numbers.
The first number is 2n, the second 2n + 2, and the third is 2n + 4
2n + 2n + 2 + 2n + 4 = 6n + 6 = 240
6n = 234, n = 36, numbers are 72, 74, 76
Example
In a triangle, angle A = angle B and Angle C is 42 degrees less
than angle A. Find the angles
Solution
In a triangle, angle A = angle B and Angle C is 42 degrees less
than angle A. Find the angles
Angle A = Angle B = x. Angle c = x – 42
x + x + x – 42 = 180, 3x = 222, x = 74 deg
Example
• A taxi costs $3.00 plus $0.80 cents a mile. How far can you
travel for $7.80?
Solution
• A taxi costs $3.00 plus $0.80 cents a mile. How far can you
travel for $7.80?
• Let miles = m, cost = 3 + 0.8m = 7.8
• 0.8m = 4.8, m = 6 miles
Example
• An 18 foot wire is to be cut so that the length of the longer
piece is 5 times the length of the shorter piece. Find the
length of each piece.
Solution
• Let x be the shorter piece, y the longer
• 18 = x + y
• y = 5x
• 18 = x + 5x = 6x
x = 3 ft, the shorter piece. The longer is 5x = 15 ft
Example
• Through the year 2010 California will have 21 more electoral
votes for president than Texas. If the total votes of the two
states is 80, find the votes for each state
Solution
• T + C = 89
• C = T + 21
T + C = T + T + 21 = 2T + 21 = 89
So 2T = 89 – 21 = 68, T = 34, C = 55
Example
• A car rental company charges $28 a day and $0.15 per mile.
If you rent a car for a day and your bill is $52, how far did you
drive?
Solution
• A car rental company charges $28 a day and $0.15 per mile.
If you rent a car for a day and your bill is $52, how far did you
drive?
• Your bill is $52. Of that $28 is for the day. This leaves
52 – 28 = 24 for mileage. Let M be the number of miles
0.15 x M = 24
M = 24/0.15 = 2400/15 = 160 miles
Section 9.5
Formulas and Problem Solving
Formulas
Formulas give relationships between quantities
– Distance = rate times time
– Perimeter of a rectangle = twice the length plus twice
the width
– The area of a rectangle is length times width
– Area of a circle = π r2
Example
• If we can travel 5 miles per hour and we have traveled for four
hours, how far did we go?
Solution
• If we can travel 5 miles per hour and we have traveled for four
hours, how far did we go?
•
Distance = speed x time
• D = 5 mph x 4 hrs = 20 mi
Example
• A family is planning a vacation drive that is 1180 miles long.
• If they drive 50 mph, how long will it take them to reach their
goal?
solution
• A family is planning a vacation drive that is 1180 miles long.
• If they drive 50 mph, how long will it take them to reach their
goal?
• Distance = speed times time. = D
• Let H be the number of hours
• 1180 =H (speed) = 50 H
• H = 1180/50 = 96 miles
Example
• Tom has enough fence for a rectangular garden with
perimeter 140 feet. If his garden is 30 ft wide, how long can it
be?
Solution
• Tom has enough fence for a rectangular garden with
perimeter 140 feet. If his garden is 30 ft wide, how long can it
be?
• Perimeter = 2L + 2W, where L and W are length and width
• 140 = 2L + 2(30)
• 140 = 2L + 60
• 80 = 2L
• L = 40 feet long
Section 9.5
Formulas
Solving for a Variable
Distance = rate times time: D=rt Solve for t:
Need to isolate t. Divide both sides by r:
t = D/r
Volume: the volume of a box is lwh, V = lwh, where l is length, w
is width, and h is height. Solve for h
Need to isolate h. Divide both sides by lw
h = V/lw
Example
The perimeter of a rectangle, P = 2l + 2w, where l is the length
and w is the width. Solve for w
Solution
P = 2l + 2w
subtract 2l from each side
P – 2l = 2w
divide both sides by 2
w = (P – 2l) / 2
Example
Temperature conversion
F = 9/5 C + 32
Solve for C
Solution
Temperature conversion
F = 9/5 C + 32
Solve for C
Subtract 32 from both sides:
F – 32 = 9/5 C
Multiply both sides by 5/9
(5/9)(F – 32) = C
Solving for a Variable
• Solve T = mnr for n
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Solution
• Solve T = mnr for n
• n = T / (mr)
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Example
• Solve for A = PRT for T
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Solution
• Solve for A = PRT for T
• Divide by PR, A/PR = T
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Section 9.6
Percent and Mixture Problems
Example
• The number 63 is what percent of 72?
Solution
• 63 = n% x 72
• 63/72 = n%
• n = 0.875
• 87.5 %
Example
• A realtor earns 4% commission. If she earned $6000 on the
last house she sold, what was the sales price of the house?
Solution
• A realtor earns 4% commission. If she earned $6000 on the
last house she sold, what was the sales price of the house?
• 6000 = 4% x Price
• 6000/0.04 = 600,000/ 4 = 150,000 = price
Example
• Mark is taking Peggy out to dinner. He has $66 to spend. If
he wants to tip the server 20%, how much can he afford to
spend on the meal?
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Solution
• Mark is taking Peggy out to dinner. He has $66 to spend. If
he wants to tip the server 20%, how much can he afford to
spend on the meal?
• Cost = base + tip x base
• Let B = Base
• 66 = B + 0.2 B = 1.2B
• B = 66/1.2 = 55,
• He can spend $55
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Example
• The cost of a certain car increased from $16,000 last year to
$17,280 this year. What was the percent of increase?
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Solution
• The cost of a certain car increased from $16,000 last year to
$17,280 this year. What was the percent of increase?
• Increase = n% ( cost)
• Cost this year = 17280
• Cost last year = 16000
• Increase = 17280 – 16000 = 1280
• 1280 = n% (16000)
• n% = 1280/16000 = 128/1600 = 0.08 or 8%
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Example
• Patrick weighed 285 pounds two years ago. After dieting, he
reduced his weight to 171 pounds. What was the percent of
decrease in his weight?
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Solution
• Patrick weighed 285 pounds two years ago. After dieting, he
reduced his weight to 171 pounds. What was the percent of
decrease in his weight?
• Decrease = 285 – 171 = 114
• Decrease = n% ( 285)
• 114 = n% (285)
• n% = 114/285 = 0.4 or 40%
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Section 9.7
Solving Inequalities
Inequalities
Represents the set {xx ≤ 7}
7
Represents the set {xx > – 4}
-4
63
Example
Graph: −2 < x ≤ 5
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Additive Property
• If a < b then a + c < b + c
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Example
2x – 4 < x + 6
+4
+4 add 4
2x < x + 10
-x
-x
subtract x
x < 10
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Multiplicative Property
• If a < b then
ca < bc if c > 0
ca > bc if c < 0
• When we multiply or divide by a negative number, we change
the sign
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Example
2x – 4 < x + 6
+4
+4 add 4
2x < x + 10
-x
-x
2x – 4 < x + 6
-6
- 6 subtract 6
2x – 10 < x
subtract x
x < 10
or -x > -10
-2x
- 2x
subtract 2x
-10 < -x
or 10 > x
We used only addition, but proved the multiplication rule.
If we multiply or divide by -1, we change the signs and
change the inequality
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Inequalities vs. Equalities
• An equality has a equal sign: Ax + B = C
• An inequality has one of the following:
– Greater than or equal to: ≥
– Less than or equal to: ≤
– Greater than: >
– Less than: <
• We use the same techniques to solve them, but the answers
are quite different
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Examples of Types of Expressions
a) 4x – 5
Neither an equation nor an inequality
b) 4x – 5 = 1
An equation or equality
c) 4x – 5 < 11
A linear inequality in one variable
d) 4x2 – 5 < 11
An inequality, but not linear
e) 4x – 5y < 3
A linear inequality in two variables
More Complex Inequalities
Do the following satisfy 6x – 2 < 5x – 4?
• x = -3
• x = -1.5
Solution
Do the following satisfy 6x – 2 < 5x – 4?
Substitute values
• x = -3
6( -3) – 2 < 5(-3) – 4
-18 – 2 < -15 – 4, or – 20 < - 19
YES
• x = -1.5
6(-1.5) -2 < 5(-1.5) – 4
-9 – 2 < - 7.5 – 4
-11 < - 11.5
NO
Examples
Find the solution of (values of x for which it is true) and graph:
6v – 2 < 5v - 3
Solution
6v – 2 < 5v – 3
Add -5v to both sides
6v – 5v – 2 = 5v – 5v – 3
v – 2 < -3
Add 2 to both sides
v < 2 – 3 = -1
v < -1
Parentheses?
Just distribute as always:
3(x – 4) < -2
3x – 12 < -2
3x < 10
x < 10/3
Example
Is x = 2 a solution to the following:
• x>3
• x ≥ -1
• 3x – 5 ≥ 2(x – 1)
• 5x – 3 ≤ 2x - 3
Solution
Is x = 2 a solution to the following:
• x > 3; NO
• x ≥ -1; YES
• 3x – 5 ≥ 2(x – 1) or 3x – 5 ≥ 2x – 2:
subtract 2x from both sides: 5x – 5 ≥ -2
add 5 to both sides: 5x ≥ 3; YES
• 5x – 3 ≤ 2x – 3
subtract 2x from both sides, 3x – 3 ≤ – 3
add 3 to both sides: 3x ≤ 0; NO
Words to Math
• x is at least 4:
• x is at most 4:
• x exceeds 2:
• x never exceeds 2:
• x is a minimum of 2:
Solution
• x is at least 4: x ≥ 4
• x is at most 4: x ≤ 4
• x exceeds 2: x > 2
• x never exceeds 2: x ≤ 2
• x is a minimum of 2: x ≥ 2
Example
Four times the quantity, x plus three, exceeds eleven more than
3 times x
Solution
Four times the quantity x plus three exceeds eleven more than
3 times x
4 (x + 3) > 11 + 3x
4x + 12 > 3x + 11
4x – 3x + 12 > 3x – 3x + 11
x + 12 > 11
x + 12 – 12 > 11 – 12
x > -1
Examples
Six, plus four times x, is greater than or equal to three times x,
plus five
Solution
Six plus four times x is greater than or equal to three times x,
plus five
6 + 4x ≥ 3x + 5
x ≥ -1
Example
• Five, minus two times x, is less than or equal to three times x,
plus 5
Solution
Five minus two times x is less than or equal to three, times x
plus 5
5 – 2x ≤ 3 (x+5)
5 – 2x ≤ 3 x + 15
-10 ≤ x
Example
• Five times w never exceeds three more than four times w
Solution
• Five times w never exceeds three more than four times w
5w ≤ 4w + 3
w≤3
Solve
• 5 (w-2) + 14 > 4 (w+3)
Solution
• 5 (w-2) + 14 > 4 (w+3)
5w – 10 + 14 > 4w + 12
5w + 4 > 4w + 12
w>8
Solve
• 5(2m + 8) + 3m > 3 (4m + 5) - 4
Solution
• 5(2m + 8) + 3m > 3 (4m + 5) - 4
10m + 40 + 3m > 12m + 15 – 4
13m + 40 > 12m + 11
m > - 29
Example
• You are having a catered event. You can spend at most
$1200. The set up fee is $250 plus $15 per person, find
the greatest number of people that can be invited and
still stay within the budget.
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Solution
• You are having a catered event. You can spend at most
$1200. The set up fee is $250 plus $15 per person, find
the greatest number of people that can be invited and
still stay within the budget.
• 1200≥ 250 + 15 n, where n is the number of people
• Subtract 20 from each side
• 1950 ≥ 15 n
• 63.3 ≥ n
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Example
The daily cost of producing x units of computer mice includes a
fixed cost of $450 per day and a variable cost of $4 per unit.
The income produced by selling x units is $5 per unit.
Determine the values of x for which the company earns a
daily profit.
Solution
The daily cost of producing x units of computer mice includes a
fixed cost of $450 per day and a variable cost of $4 per unit.
The income produced by selling x units is $5 per unit.
Determine the values of x for which the company earns a
daily profit.
Cost = fixed cost + variable cost times the number of units
Profit = income – cost
Cost = 450 + 4x, where x is the number of units
Income = 5x, where x is the number of units
Profit = 5x – 450 – 4x = x – 450
x – 450 > 0, or x > 450 units
Example
To earn an A in algebra, a student must have at least 630 points.
He has 535 points going into the last, 100 point test. How
many points must he score on the last test?
Solution
To earn an A in algebra, a student must have at least 630 points.
He has 535 points going into the last, 100 point test. How
many points must he score on the last test?
535 + x > 630
x > 95
Example
• A farmer has 84 ft of fence that he wants to use to fence three
sides of a rectangular pen with sides 12ft, x ft, and 12 ft. What
is the largest area he can fence in?
Solution
• A farmer has 84 ft of fence that he wants to use to fence three
sides of a rectangular pen with sides 12ft, x ft, and 12 ft. What
is the largest area he can fence in?
Fencing needed is 12 + x + 12. This must be less than or equal
to 84 ft
12 + x + 12 ≤ 84
x + 24 ≤ 84
x ≤ 60
Area is length times width, = 12 x. The area ≤ 60 (12) = 720 ft
Example
1. The sum of any two sides of a triangle is greater than the
length of the third sides. If the lengths of the sides are a, b,
and c, find three different inequalities illustrating the
statement
2. Write an expression for:
a. 3 more than x
b. 3 is more than x
c. 2 less than x
d. 2 is less than x
Solution
1. The sum of any two sides of a triangle is greater than the
length of the third sides. If the lengths of the sides are a, b,
and c, find three different inequalities illustrating the
statement
a + b > c, a + c > b, c + b > a
2. Write an expression for:
a. 3 more than x: x + 3
b. 3 is more than x: 3 > x
c. 2 less than x: x - 2
d. 2 is less than x: 2 < x