College Algebra

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For example, to solve the inequality x2 – 5x ≤ –6

We first move all the terms to the left-hand side and factor to get (x – 2)(x – 3) ≤ 0

This form of the inequality says that the product (x – 2)(x – 3) must be negative or zero.

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Solving Nonlinear Inequalities

So to solve the inequality, we must determine where each factor is negative or positive.

This is because the sign of a product depends on the sign of the factors.

The details are explained in Example 3, in which we use the following guidelines.

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Guidelines for Solving Nonlinear Inequalities

Example 3 illustrates the following guidelines for solving an inequality that can be factored.

Move all terms to one side.

Factor.

Find the intervals.

Make a table or diagram.

Solve.

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Guideline 1 for Solving Nonlinear Inequalities

Move all terms to one side.

If necessary, rewrite the inequality so that all nonzero terms appear on one side of the inequality sign.

If the nonzero side of the inequality involves quotients, bring them to a common denominator.

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Guideline 2 for Solving Nonlinear Inequalities

Factor.

Factor the nonzero side of the inequality.

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Guideline 3 for Solving Nonlinear Inequalities

Find the intervals.

Determine the values for which each factor is zero.

These numbers will divide the real line into intervals.

List the intervals determined by these numbers.

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Guideline 4 for Solving Nonlinear Inequalities

Make a table or diagram.

Use test values to make a table or diagram of the signs of each factor on each interval.

In the last row of the table, determine the sign of the product (or quotient) of these factors.

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Guideline 5 for Solving Nonlinear Inequalities

Solve.

Determine the solution of the inequality from the last row of the sign table.

Be sure to check whether the inequality is satisfied by some or all of the endpoints of the intervals.

This may happen if the inequality involves ≤ or ≥.

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Guidelines for Solving Nonlinear Inequalities

The factoring technique described in these guidelines works only if all nonzero terms appear on one side of the inequality symbol.