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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 82
Chapter 2, Problem 82

Solve each rational inequality. Give the solution set in interval notation.
(9x−8)(4x2+25)<0\(\frac{(9x-8)}{(4x^2+25)}\)<0

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1
Identify the rational inequality to solve: \(\frac{9x - 8}{4x^2 + 25} < 0\).
Determine the critical points by setting the numerator and denominator equal to zero separately: solve \(9x - 8 = 0\) and \(4x^2 + 25 = 0\).
Note that \(4x^2 + 25 = 0\) has no real solutions because \(4x^2 = -25\) is impossible for real \(x\), so the denominator is always positive.
Since the denominator is always positive, the sign of the rational expression depends solely on the numerator \(9x - 8\); find where \(9x - 8 < 0\).
Solve \(9x - 8 < 0\) to find the solution set, then express the solution in interval notation.

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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 this ratio to zero or another value. Solving them requires understanding where the expression is positive, negative, or zero by analyzing the signs of numerator and denominator.
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Sign Analysis and Critical Points

To solve rational inequalities, identify critical points where the numerator or denominator equals zero. These points divide the number line into intervals. Testing each interval determines where the inequality holds true, considering that division by zero is undefined.
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Interval Notation

Interval notation is a concise way to express solution sets of inequalities. It uses parentheses for values not included (like points causing zero denominators) and brackets for included endpoints. Understanding this notation helps clearly communicate the solution.
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Related Practice
Textbook Question

For each equation, (b) solve for y in terms of x. See Example 8.

2x2+4xy−3y2=22x^2 + 4xy - 3y^2 = 2

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Solve each equation. See Example 7. (3x+7)1/3-(4x+2)1/3=0

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Textbook Question

For each equation, solve for x in terms of y. 2x2 + 4xy - 3y2 = 2

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Solve each rational inequality. Give the solution set in interval notation. (5-3x)2/(2x-5)3>0

Textbook Question

Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) x2 - 8x + 16 = 0