Algebra Demystified 2nd Ed

Chapter 9 linear inequaliTieS 321

Some word problems give two alternatives and ask for what interval of the
variable is one alternative more attractive than the other. If the alternative is
between two costs, for example, in order for the cost of A to be more attractive
than the cost of B, we solve “Cost of A < Cost of B.” If the cost of A to be no
more than the cost of B (also the cost of A to be at least as attractive as the cost
of B), we solve “Cost of A < Cost of B.” If the alternative is between two
incomes of some kind, for the income of A to be more attractive than the
income of B, solve “Income of A > Income of B.” If the income of A is to be at
least as attractive as the income of B (also the income of A to be no less attrac-
tive than the income of B), solve “Income of A > Income of B.”
Some of the following examples and practice problems are business prob-
lems, so we now review a few business formulas. Revenue is normally the price
per unit times the number of units sold. For instance, if an item sells for $3.25
each and x represents the number of units sold, the revenue is represented by
3.25x (dollars). The total cost generally includes overhead costs (sometimes
called fixed costs) and production costs (sometimes called variable costs). The
overhead cost is a fixed number (no variable). The production cost is usually
computed as the cost per unit times the number of units sold. The total cost is
usually the overhead costs plus the production costs. Profit is revenue minus
cost. If a problem asks how many units must be sold to make a profit, solve
“Revenue > Total Cost.”

EXAMPLE

A manufacturing plant, which produces compact disks, has monthly over-

head costs of $6,000. Each disk costs 18 cents to produce and sells for 30 cents.

How many disks must be sold in order for the plant to make a profit?

Let x = number of CDs produced and sold monthly

In order for the plant to make a profit, revenue must be more than cost, so

we need to solve the inequality Revenue > Cost, where Cost = 6000 + 0.18x

and Revenue = 0.30x.

Revenue > Cost

030 6000 018

018018

012 6000

.   .
. .
. 

xx

xx

x

>  + 

–> –

> 

xx

x

.

 ,

> 

> 

6000

012

50 000

EXAMPLE

A manufacturing plant, which produces compact disks, has monthly over-