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Applications of Linear and Quadratic Equations
Solving Verbal Problems
 Read the question carefully.
 Choose a letter to represent the unknown
quantity.
 Use facts and relationships from the problem to
create equation(s) involving the letter.
 Solve the equation.
 Check the solution to see if it answers the
question.
 Interpret and then explain the meaning of the
solution found.
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Basic Relationships
profit = total revenue - total cost
total revenue = (price per unit)(number of units sold)
=pq
total cost = variable cost + fixed cost
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Example 1
A person invested $20 000: part at an interest rate
of 6% annually and the remainder at 7% annually.
The total interest at the end of 1 year was
equivalent to an annual 6 ¾ % rate on the entire
$20 000. How much was invested at each rate?
Simple Interest = Principal.rate.time
SI = P×r×t
Let the value invested at 6% be x. Then the
amount invested at 7% is (20 000 - x).
3
Looking at the interest earned in 1 year:
x(.06)(1)  (20000  x)(.07)(1)  20000(.0675)(1)
0.06 x  20000 (0.07)  0.07 x  1350
 0.01x  1400  1350
1350  1400
x
 0.01
 5000
$5000 invested at 6 % and $15000 invested at 7%.
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Example 2
The cost of a product to a retailer is $3.40. If
the retailer wishes to make a profit of 20% on
the selling price, at what price should the
product be sold?
Let the selling price be x.
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Cost = $3.40
Profit = 20% of selling price = 0.2x
3.40  0.2 x  x
3.40  x  0.2 x
3.40  0.8 x
3.40
x
0 .8
4.25  x
The product should be sold for $4.25.
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Example 3
A college dormitory houses 210 students. This
fall (autumn), rooms are available for 76
freshman (first-years). On the average, 95% of
those freshman who request room applications
actually reserve a room. How many room
applications should the college send out if it
wants to receive 76 reservations?
Let the number of applications be x.
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95% of x  76
0.95 x  76
76
x
0.95
x  80
The college should forward 80 applications
for rooms.
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Example 4
A compensating balance refers to that practice
wherein a bank requires a borrower to maintain on
deposit a certain portion of a loan during the term
of the loan. For example, if a firm makes a
$100000 loan which requires a compensating
balance of 20%, it would have to leave $20000 on
deposit and would have the use of $80000.
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To meet the expenses of retooling, the Victor
Manufacturing Company must borrow $95000.
The Third National Bank, with whom they have
had no prior association, requires a
compensating balance of 15%. To the nearest
thousand dollars, what must be the amount of
the loan to obtain the needed funds?
Let the amount to be borrowed be x.
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If 15% is the compensating balance, then
$95000 must represent 85% of the loan.
85% of x  $95 000
0.85 x  95 000
95 000
x
0.85
x  111 764 .11
x  112 000 nearest '000
The company should borrow $112 000.
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Inequalities
An inequality is a statement about the order
of two numbers. That is, that one number is
less than another.
Symbols to show this order are:




less than
less thanor equal to
greater than
greater than or equal to
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Examples
x7
7
)
4
x  4
2 x6

2

6
)
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Solving Inequalities
Care must be taken to maintain the sense of the
inequality.
The allowable operations are the same as those for
equations with two exceptions.
1. When you multiply both sides of the inequality
by a negative number the sense is reversed.
2. When finding the reciprocal of each side, the
sense is reversed.
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Example
2  4  2  4 ???
No,  2  4
So,  x  t  x  t
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Example
1 1
2  4   ??
2 4
1 1
No, 
2 4
1 1
So,   x  2
x 2
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Example
3(2t  2) 6t  3 t


2
5
10
15(2t  2)  2(6t  3)  t
30t  30  12t  6  t
30t  12t  t  6  30
17t  24
24
t
17
(
24
17
t
17
Example
5 y  1 7( y  1)

3
2
 7( y  1) 
5 y  1  3 
 2 
 2(5 y  1)  37( y  1)
10 y  2  21 y  21
21y  10 y  21  2
11 y  23
23
y
11
y

)
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11
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Example
A company produces alarm clocks. During the
regular work week, the labour cost for producing
one clock is $2.00. However, if a clock is
produced in overtime the labour cost is $3.00.
Management has decided to spend no more than a
total of $25000 per week for labour. The
company must produce 11000 clocks this week.
What is the minimum number of clocks that must
be produced during the regular work week?
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Let the number of clocks that must be produced
in a regular week be x.
Then the number produced in overtime is
11000 - x.
2 x  3(11000 x)  25000
2 x  33000  3x  25000
 x  8000
x  8000
The minimum number to be produced is 8000.
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Example
The current ratio of Precision Machine Products
is 3.8. If their current assets are $570 000, what
are their current liabilities? To raise additional
funds, what is the maximum amount they can
borrow on a short-term basis if they want their
current ratio to be no less than 2.6?
(See Example 3 in Section 1.3 of the textbook for
an explanation of current ratio).
21
Current ratio =
assets
liabilities
570 000
3 .8 
x
570 000
x
3.8
 150 000
Current liabilities are $150 000.
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570 000  x
 2.6
150 000  x
570000  x  2.6(150000  x)
570 000  x  390 000  2.6 x
570 000  390 000  2.6 x  x
180 000  1.6 x
112 500  x
The company can borrow at most $112 500 on a
short term basis.
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Absolute Value
The absolute value of a number can be thought
of as its distance from zero.
eg 4 and -4 are both 4 units from the origin.
They both have an absolute value of 4.
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Absolute Value
Note:
1. The absolute value is always positive.
2.
a b  ba
3.
ab  a b
4.
a
a

b
b
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Example
-8
0
8
x-5
x-5
x 5  8
x  5  8 or x  5  8
x  13
or x  3
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Equations and Inequalities Involving Absolute Value
Example
x7  2
-2
(
0
2
)
x+7
2 x7 2
27  x  27
 9  x  5
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Example
3x  8
4
2
-4

0
3x  8
2
4

3x  8
2
3x  8
3x  8
 4 or
4
2
2
3x  8  8 or 3x  8  8
3x  0
or
x0
or
3x  16
16
x
3
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