Money, Mixture, Motion, and Inequalities

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.

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).

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:

1. Simplify both sides (distribute and combine like terms).

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

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

4. 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

1.  3x - 5  >  6(x - 1) 2. 5(x + 2)  <  2(x - 3) Back to Math 152A Home Page

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