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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Solving Inequalities

A real number a is a solution of an inequality if the inequality is satisfied (is true) when a is substituted for x. The set of all solutions is the solution set of the inequality.

Example 1

Solving an Inequality

Solve 2x - 5 < 7.

Solution

 2x - 5 < 7 Original inequality 2x - 5 + 5 < 7 + 5 Add 5 to both sides. 2x < 12 Simplify. Multiply both sides by x < 6 Simplify.

The solution set is (-∞, 6).

NOTE In Example 1, all five inequalities listed as steps in the solution are called equivalent because they have the same solution set.

Once you have solved an inequality, check some x-values in your solution set to verify that they satisfy the original inequality. You should also check some values outside your solution set to verify that they do not satisfy the inequality. For example, the figure below shows that whenx = 0 or x = 5 the inequality 2x - 5 < 7 is satisfied, but when x = 7 the inequality 2x - 5 < 7 is not satisfied.

Example 2

Solving a Double Inequality

Solve -3 ≤ 2 - 5x ≤ 12.

Solution

 -3 ≤ 2 - 5x ≤ 12 Original inequality -3 - 2 ≤ 2 - 5x - 2 ≤ 12 - 2 Subtract 2. -5 ≤ - 5x ≤ 10 Simplify. Divide by -5 and reverse both inequalities. 1 ≥ x ≥ -2 Simplify.

The solution set is [-2, 1], as shown in the figure below.

The inequalities in Examples 1 and 2 are linear inequalitiesâ€”that is, they involve first-degree polynomials.