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 Algebra: Inequalities

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 a profit.

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 equations.

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.

 Solving InequalitiesView animation

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?

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