Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 68

Chapter 2, Problem 68

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

Verified step by step guidance

Start by rewriting the inequality to have zero on one side: $\frac{3}{x+5} - 2 > 0$.

Find a common denominator to combine the terms: $\frac{3}{x+5} - \frac{2(x+5)}{x+5} > 0$.

Simplify the numerator: $\frac{3 - 2(x+5)}{x+5} > 0$, which becomes $\frac{3 - 2x - 10}{x+5} > 0$ or $\frac{-2x - 7}{x+5} > 0$.

Identify critical points by setting numerator and denominator equal to zero: numerator $-2x - 7 = 0$ and denominator $x + 5 = 0$. Solve these to find $x = -\frac{7}{2}$ and $x = -5$.

Use these critical points to divide the number line into intervals, then test a value from each interval in the inequality $\frac{-2x - 7}{x+5} > 0$ to determine where the inequality holds true. Remember to exclude $x = -5$ since it makes the denominator zero.

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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 side is a ratio of polynomials. Solving them requires finding values of the variable that make the inequality true, considering where the expression is defined and the sign of the numerator and denominator.

Domain Restrictions

The domain of a rational expression excludes values that make the denominator zero, as division by zero is undefined. Identifying these restrictions is crucial before solving inequalities to avoid invalid solutions.

Interval Notation and Sign Analysis

After determining critical points from numerator and denominator, the number line is divided into intervals. Testing each interval helps determine where the inequality holds. Solutions are then expressed in interval notation, clearly showing the valid ranges.

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Interval Notation

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