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 56

Chapter 4, Problem 56

Solve each rational inequality. Give the solution set in interval notation. (3x + 7)/(x - 3) ≤ 0

Verified step by step guidance

Identify the critical points by setting the numerator and denominator equal to zero separately: solve $3x + 7 = 0$ and $x - 3 = 0$. These points divide the number line into intervals.

Determine the intervals based on the critical points found: one interval to the left of the smaller critical point, one between the two critical points, and one to the right of the larger critical point.

Test a sample value from each interval in the inequality $\frac{3x + 7}{x - 3} \leq 0$ to check whether the expression is less than or equal to zero in that interval.

Include the points where the numerator is zero in the solution set because the expression equals zero there, but exclude points where the denominator is zero since the expression is undefined at those points.

Combine the intervals where the inequality holds true and express the solution set in interval notation, carefully considering whether endpoints are included or excluded.

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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 numerator and denominator separately.

Critical Points and Sign Analysis

Critical points are values that make the numerator or denominator zero, dividing the number line into intervals. By testing values in each interval, you determine the sign of the rational expression, which helps identify where the inequality holds true.

Interval Notation and Domain Restrictions

Interval notation expresses solution sets compactly using parentheses and brackets to indicate open or closed intervals. Domain restrictions exclude values that make the denominator zero, ensuring the solution set only includes valid inputs for the rational expression.

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Interval Notation

Related Practice

Textbook Question

Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. Zero of -3 having multiplicity 3; ƒ(3)=36

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Textbook Question

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

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Textbook Question

Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. Zeros of -2, 1, and 0; ƒ(-1)=-1

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=x⁴-x³+3x²-8x+8; no real zero greater than 2

Textbook Question

For each polynomial function, identify its graph from choices A–F.

$ƒ (x) = (x - 2) (x - 5)$

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) = 5x⁴ + 2x³ -x+3; k=2/5