Money, Mixture, Motion, and
Inequalities
Money Problems
Example
You have 40 coins in nickels and dimes. How many dimes
do you have if you have a total of $2.85?
Solution:
Our answer is
"We have ____ dimes"
Let
d = the number of dimes you
have
then
40  d =
the number of nickels that you have.
The total money that you have is
10d + 5(40  d)
= 285
Value of dimes + Value of nickels
= 285
10d + 200  5d
= 285
Distributing the
5
5d + 200 =
285
10d  5d = 5d
5d = 85
Subtracting 200 from both sides
d = 17
Dividing by
5
We have 17 dimes.
Example
You are the manager of the new Tahoe Stadium. You sell your VIP seats
for $200 each and your general admission seats for
$75. Your stadium
holds 10,000 people, and you need to earn at least $1,000,000. If
you sell out, how many of your seats should you designate as VIP seats?
Solution:
Our answer should be
"We should designate ________ as
VIP seats."
Let
x =
number of VIP seats
then
10,000  x =
number of general admission seats.
The money from the VIP seats is
200x
and the money from the general admission seats is
75 (10,000  x)
Hence
200x + 75(10,000  x) = 1,000,000
VIP
money + general Ad money = 1,000,000
200x + 750,000  75x = 1,000,000
Distributing
the 75
125x + 750,000 = 1,000,000
200x  75x = 125x
125x = 250,000
Subtracting 750,000
x = 2,000
Dividing by 125
We designate 2,000 seats as VIP seats.
Mixture Problems
Example
Vodka contains 40% alcohol and wine contains 10% alcohol. You want
to make a new drink that is 20% alcohol using vodka and wine. How much
of each should you use to make 15 ounces of this drink?
Solution:
Our answer should be
"Use _______ ounces of vodka
and _________ ounces of wine.
We let
x =
number of ounces of vodka
Then
15  x
is the number of ounces of wine
Note that the amount of alcohol in the final mixture is
15
(0.2) = 3
15 ounces times
20% alcohol =
3
Hence we can write
0.4x + 0.1(15  x)
= 3 vodka alcohol + wine alcohol = total alcohol
Multiplying by 10 to get rid of the
decimal, we get:
4x + (15  x) = 30
4x + 15  x = 30
3x = 15
4x  x = 3x and
30  15 = 15
x = 5
dividing by
3
Hence we pour 5 ounces of vodka and 10 ounces of wine. (It is not
recommended to try this at home).
Motion Problems
Example
Suppose that I am walking from school at 3 miles per hour and start at
12:00.
At 12:30, you start riding your bike at 18 miles per hour to find me.
At what time do you find me?
Solution:
The answer is
"You find me at _______"
Let
t = the time after
12:00
We use the formula
distance = rate times time
Then my distance from school is
3t
To find your distance from school, multiply the rate, 18
by the time since you left, t  1/2.
18 (t  1/2)
We set the two equal to each other:
3t =
18(t  1/2)
3t =
18t  9
distributing through
15t =
9
subtracting 18t from both sides
t =
9/15 = 3/5 = 36/60 or 36
minutes. dividing both sides by
15
Hence you find me at 12:36.
Linear Inequalities
Definition
A linear inequality is one that can be reduced
to
ax + b < 0
or
ax + b > 0
or
ax + b < 0
or
ax + b > 0 
Step by step method for solving linear inequalities:

Simplify both sides (distribute and combine like terms).

Bring the x's to the left and the constants to the right.

Divide by the coefficient (changing the inequality if the sign of
the inequality is negative).

Plot on a number line (remember holes and dots).
Example
Solve
2(x  9) <
3(2x  10)
Solution
2x  18 <
6x  30
Distributing the
2 and the
3
4x <
12
Subtracting
6x and adding
18
x >
3
Dividing by
4
Plot on a number line with a hole at 3 and an arrow to the right of
3.
To play with the number line go to
Inequality
Play
Exercises

3x  5 > 6(x  1)

5(x + 2) < 2(x  3)
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