Introduction to Linear Inequality Word Problems
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.
A < B |
A > B |
A is less than B |
A is greater than B |
A is smaller than B |
A is larger than B |
B is greater than A |
B is less than A |
B is larger than A |
A is more than B |
A ≤ B |
B ≤ A |
A is less than or equal to B |
B is less than or equal to A |
A is not more than B |
B is not more than A |
B is at least A |
A is at least B |
B is A or more |
A is B or more |
A is no greater than B |
B is no greater than A |
B is no less than A |
A is no less than B |
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.”
Linear Inequalities for Business Word Problems
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 3.25 x (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.”
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