An inequality is a mathematical statement that says two
expressions are not necessarily equal, and describes the relationship
of one expression to the other. Inequalities could be used to describe
the fact that employees need to work 40 or fewer hours a week. A
company could also employ an inequality to indicate that its revenues
must be greater than a certain amount in order for the company to make
Like equations, inequalities are written using combinations of
constants, variables, coefficients, and mathematical operators.
However, instead of using the equality symbol (=), inequalities use the
following comparison symbols:
The solution to an inequality differs from that of an equation in
one significant way: An inequality's solution is defined in terms of intervals,
while an equation's solution is defined as a specific number. Consider
the graphs below. The solution to the inequality is the interval of
numbers that is equal to or less than 3. In the equation, the solution
is only 3.
Inequalities are solved the same way equations are, with one major
exception: When you multiply or divide by a negative number, reverse
the direction of the inequality symbol.
Consider the following inequality problem. Your company's workers'
compensation insurance policy features a $5,000 deductible after which
90 percent of the eligible expenses are reimbursed by the insurance
company. Your company is investigating the purchase of a "stop loss"
policy, which would cover the amount not paid for by the first policy
up to a maximum of $20,000 per claim. As the head of the human
resources department, you must determine the maximum amount of a claim
that will be paid for entirely by the two insurance policies.
First, let x = amount of a claim. The amount not covered by the first policy is the deductible ($5,000) plus 10 percent of the claim (x).
The amount covered by the stop loss policy should not exceed $20,000.
Set up the inequality and solve using the same steps used for solving
The company will not have to pay any part of claims less than $150,000.
View the animation below for a further illustration of inequalities used in a retail scenario.
1. Solve for y.
y + 9 < 11
2. Solve for x.
3. Solve for x.
4. A company has costs of $50 per unit and $1,000 in overhead per
month. The company does not want to exceed $60,000 in expenditures per
month. What is the interval on which it should produce?
5. A company sells its product for $30 and has costs of $10 per unit
and $400 in overhead per month. If the company wishes to turn a profit
each month, on what interval should it produce?