Applications
Linear inequality word problems are solved much the same way as linear equality word problems. There are two important differences. Multiplying and dividing both sides of an inequality by a negative quantity requires that the sign reverse. You must also decide which inequality sign to use: <, >, ≤, and ≥. The following tables should help.
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, 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), 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
attractive than the income of B), solve ‘‘Income of A ≥ Income of B.’’
Some of the following examples and practice
problems are business problems.
Let us
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). Cost tends to consist of overhead costs (sometimes called fixed
costs) and production costs (sometimes called variable costs). The overhead costs
will be a fixed number (no variable). The production costs 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.’’
Examples
A
manufacturing plant, which produces compact disks, has monthly overhead costs
of $6000. 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
Cost = 6000 +
0.18x and Revenue = 0.30x
Revenue >
Cost
The plant
should produce and sell more than 50,000 CDs per month in order to make a
profit.
Mary
inherited $16,000 and will deposit it into two accounts, one paying 5 ½ % interest
and the other paying 6 ¾ % interest. What is the most she can deposit into the
5 ½ % account so that her interest at the end of a year will be at least $960?
Let x = amount
deposited in the 5 ½ % account 0.055x = interest earned at 5 ½ %
16,000 – x =
amount deposited in the 6 ¾ % account
0.0675(16,000
– x) = interest earned at 6 ¾ % Interest earned at 5 ½ %þ Interest earned at 6 ¾
% ≥ 960.
Mary can invest no more than $9600 in the 5 ½
% account in order to receive at least $960 interest at the end of the year.
An
excavating company can rent a piece of equipment for $45,000 per year. The
company could purchase the equipment for monthly costs of $2500 plus $20 for
each hour it is used How many hours per year must the equipment be used to
justify purchasing it rather than renting it?
Let x = number of hours per year the
equipment is used The monthly purchase costs amount to 12(2500) = 30,000
dollars annually. The annual purchase cost is 30,000 + 20x.
The equipment should be used less than 750
hours annually to justify purchasing it rather than renting it.
An amusement
park sells an unlimited season pass for $240. A daily ticket sells for $36. How
many times would a customer need to use the ticket in order for the season
ticket to cost less than purchasing daily tickets?
Let x = number
of daily tickets purchased per season
36x = daily ticket cost
Season
ticket cost < daily ticket cost
A customer would need to use the ticket more
than 6 ⅔ times (or 7 or more times) in order for the season ticket to cost less
than purchasing daily tickets.
Bank A
offers a 6 ½ % certificate of deposit and Bank B offers a 5 ¾ % certificate of
deposit but will give a $25 bonus at the end of the year. What is the least
amount a customer would need to deposit at Bank A to make Bank A’s offer no
less attractive than Bank B’s offer?
Let x = amount to deposit
If x dollars is deposited at Bank A, the
interest at the end of the year would be 0.065x. If x dollars is deposited at
Bank B, the interest at the end of the year would be 0.0575x. The total income
from Bank B would be 25 + 0.0575x.
A customer would need to deposit at least $3333.33 in Bank A to earn no less than would be earned at Bank B.
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