Formulas and Absolute Value Inequalities
Problem Solving With Formulas
A formula is an equation
that relates real world quantities.
Examples
P = 2l + 2w
is the formula for the perimeter P of a rectangle
given the length l and width w.
d = rt
is the formula for the distance traveled d given
the speed s and the time t.
A = P + Prt
is the formula for the amount A in a bank account t
years after P dollars is put in at an interest rate
of r.
V = pr^{2}h
is the formula for the volume V of a cylinder of
radius r and height h, where p
@ 3.14.
h
A =
(b_{1} + b_{2})
2
is the formula for the area A of a trapazoid with
height h and bases b_{1} and b_{2}.
We say that a formula is solved
for a variable x if the equation becomes
x = stuff
where the left hand side does not include any x's.
Example
Solve
C = 2pr
for r then determine the radius of a circle with
circumference 4.
Solution
Divide both sides by 2p:
C
= r
2p
Use the reflexive property to get
C
r =
2p
Now plug in 4 for C to
obtain
4
2
r =
=
2p
p
Steps for Solving a Word Problem

Read the problem,
sketch the proper picture, and label variables.

Write down what the
answer should look like.

Come up with the
appropriate formula.

Solve for the needed
variable.

Plug in the known
numbers.

Answer the question.

Example:
A pile of sand has the shape of a right circular cone. Find the height of
the pile if it contains 100 cc of sand and the
radius is 5 cm.
Solution:


The height of the pile is ________ cm.

We use the formula for the volume of a right circular cone:
V
= 1/3 pr^{2}h.

Multiply by 3 on both sides to
get
3V = pr^{2}h.
Divide both sides by pr^{2}
to obtain
3V
3V
= h or
h =
pr^{2}
pr^{2}

3(100)
12
h =
=
p5^{2}
p

The height of the pile is 12/p
cm.
Absolute Value Inequalities
Step by Step:
Step 1: Solve as an equality.
Step 2: Plot the points above the number line.
Step 3: If the relation is
< then include the middle portion.
If the relation is > include the outside ends.
Step 4: Graph on the number line remembering to put an open
or closed dot when necessary.
Example:

Graph the inequality
x < 4
We proceed as follows:

We have x = 4 or
x = 4.

Since the relation is "<" we include the middle portion and put
open circles.

Graph the inequality
2x +
4 > 6

We have
2x + 4 = 6
or 2x + 4 = 6

2x = 2 or
2x = 10
so
x =
1 or
x = 5
We graph the solution on a number line including the outer regions and putting
a closed dot at the endpoints.
Exercises Graph the solution set of the following.
2x + 1 >
3
3x  2 >
4
2x  1 >
2
 5x + 4 <
3
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