Ch. 3 - Polynomial and Rational Functions

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

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 60

Chapter 4, Problem 60

Solve each rational inequality. Give the solution set in interval notation. (2x - 3)/(x + 1) > 4

Verified step by step guidance

Start by rewriting the inequality to have zero on one side: $ \frac{2x - 3}{x + 1} - 4 > 0 $.

Find a common denominator and combine the terms into a single rational expression: $ \frac{2x - 3 - 4(x + 1)}{x + 1} > 0 $.

Simplify the numerator: $ 2x - 3 - 4x - 4 = -2x - 7 $, so the inequality becomes $ \frac{-2x - 7}{x + 1} > 0 $.

Determine the critical points by setting the numerator and denominator equal to zero: $ -2x - 7 = 0 $ and $ x + 1 = 0 $. Solve these to find $ x = -\frac{7}{2} $ and $ x = -1 $.

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 + 1} > 0 $ to determine where the inequality holds true. Express the solution set in interval notation, remembering to exclude points where the denominator is 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 rational expression is compared to another using inequality symbols. Solving them requires finding values of the variable that make the inequality true, often by analyzing the sign of the numerator and denominator separately.

Critical Points and Sign Analysis

Critical points are values where the numerator or denominator equals zero, dividing the number line into intervals. By testing points in each interval, you determine where the rational expression is positive or negative, which helps identify the solution set.

Interval Notation

Interval notation is a concise way to express solution sets of inequalities using parentheses and brackets to indicate open or closed intervals. It clearly shows the range of values that satisfy the inequality, excluding points where the expression is undefined.

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Interval Notation

Related Practice

Textbook Question

Find a polynomial function ƒ(x) of least degree having only real coefficients and zeros as given. Assume multiplicity 1 unless otherwise stated. 5+i and 5-i

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Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.

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Textbook Question

Solve each rational inequality. Give the solution set in interval notation. (x - 8)/(x - 4) < 3

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Textbook Question

Use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of ƒ(k). ƒ(x) = x² + 3x + 4; k = 2+i

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.

$ƒ (x) = x^{5} + 2 x^{3} - 2 x^{2} + 5 x + 5$ ; no real zero less than -1

Textbook Question

Solve each rational inequality. Give the solution set in interval notation. $\frac{x^{2} - 3 x - 4}{x^{2} + 6 x + 9} \leq 0$