5.5 Solving Rational Ineqs PDF

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Solving Rational Inequalities. Old work but useful...

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5.5 - Solving Rational Inequaliti lities es

Objectives:  Solve rational inequalities algebraically by first combining all terms into a single factored rational term using the lowest common denominator.  Identify the critical numbers of the combined rational expression as the restrictions to the denominator and the roots of the numerator.  Solve the rational inequality algebraically using a Factor Table strategy, or by using graphing technology.  As with the solution to polynomial inequalities, solving rational inequalities gives a region of possible domain values that satisfy the inequality.  Rational inequalities can be solved graphically; make sure that the vertical asymptote is not included in your solution. Example 1 - Solve the rational inequalities by graphing: (a)

12  2 2x 6

(b)

3x  2  x2 x4

 Rational inequalities can also be solved algebraically by first rearranging the inequality so that one side is zero.  We then combine the expressions by finding a common denominator. Example 2 - Combine the rational expressions by finding a common denominator. (a)

x x 3  x  2 2x  5

(b)

x 1 x 2  x x 4

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 The inequality is solved by simplifying and factoring the numerator, and then preparing a factor table consisting of the regions in between the critical numbers of the inequality (similar to what was done when solving polynomial inequalities).  The critical numbers are the roots of the numerator and the restriction values.  When making your factor table, include all factors (top and bottom) and pay attention to the sign of the inequality. Example 3 - Solve the inequalities: (a)

4 x1  2 x1

(b)

Assigned Questions: pgs. 295 - 297 # 1 - 8, 10, 11

x  8 3x  2  2x  5 x

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5.5 - Solving Rational Inequalities Example 1 - Solve the rational inequalities in the graphs: (a)

12  2 2x 6

(b)

3x  2 x x4

Example 2 - Combine the rational expressions by finding a common denominator. (a)

x x 3  x  2 2x  5

(b)

x 1 x 2  x x 4

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Example 3 - Solve the inequalities: (a)

4 x1  2 x1

(b)

Assigned Questions: pgs. 295 - 297 # 1 - 8, 10, 11

x  8 3x  2  2x  5 x...