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Ch. 1 - Equations and Inequalities
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 1 - Equations and InequalitiesProblem 86
Chapter 2, Problem 86

Solve each rational inequality. Give the solution set in interval notation. (9x-11)(2x+7)/(3x-8)3>0

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Identify the critical points by setting the numerator and denominator equal to zero separately. Solve \( (9x - 11)(2x + 7) = 0 \) and \( (3x - 8)^3 = 0 \) to find values of \( x \) where the expression is zero or undefined.
List the critical points found: the zeros of the numerator and the zero of the denominator. These points divide the number line into intervals where the expression may change sign.
Determine the sign of the expression \( \frac{(9x - 11)(2x + 7)}{(3x - 8)^3} \) on each interval by choosing a test point from each interval and substituting it into the expression. Remember that the denominator is raised to an odd power, so its sign depends on the sign of \( (3x - 8) \).
Analyze the inequality \( > 0 \) to decide which intervals satisfy the condition that the expression is positive. Exclude any points where the denominator is zero, as the expression is undefined there.
Write the solution set in interval notation by combining all intervals where the inequality holds true, making sure to use parentheses to exclude points where the expression is undefined or equals zero if the inequality is strict.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares the expression to zero or another value. Solving them requires finding where the expression is positive, negative, or zero by analyzing the signs of numerator and denominator.
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Nonlinear Inequalities

Critical Points and Sign Analysis

Critical points are values of the variable that make the numerator or denominator zero. These points divide the number line into intervals. By testing values in each interval, you determine the sign of the rational expression to find where the inequality holds true.
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Point-Slope Form

Interval Notation

Interval notation is a concise way to represent solution sets on the number line. It uses parentheses for values not included (open intervals) and brackets for values included (closed intervals). This notation clearly shows the ranges where the inequality is satisfied.
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Interval Notation
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