Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 45

Chapter 4, Problem 45

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x+3)/(x+4)<0

Verified step by step guidance

Identify the critical points by setting the numerator and denominator equal to zero separately: solve $ x + 3 = 0 $ and $ x + 4 = 0 $. These points divide the number line into intervals.

Determine the intervals created by the critical points. In this case, the critical points are $ x = -3 $ and $ x = -4 $, so the intervals are $ (-\infty, -4) $, $ (-4, -3) $, and $ (-3, \infty) $.

Test a sample value from each interval in the inequality $ \frac{x+3}{x+4} < 0 $ to check whether the expression is negative in that interval.

Based on the sign of the expression in each interval, select the intervals where the inequality holds true (where the expression is less than zero).

Express the solution set in interval notation, excluding any points where the denominator is zero (since the expression is undefined there), and then graph the solution on a real number line.

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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 zero or another expression 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 whether the rational expression is positive or negative, which helps identify where the inequality holds.

Interval Notation and Graphing Solutions

After determining the solution intervals, express them using interval notation to clearly show the range of values satisfying the inequality. Graphing on a number line visually represents these intervals, indicating included or excluded points based on inequality strictness.

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Interval Notation

Related Practice

Textbook Question

Use transformations of f(x)=1/x or f(x)=1/x² to graph each rational function. g(x)=1/(x−1)

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Solve the equation 12x³+16x²−5x−3=0 given that -3/2 is a root.

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Give the domain and the range of each quadratic function whose graph is described. The vertex is and the parabola opens up.

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Textbook Question

In Exercises 45–46, describe in words the variation shown by the given equation. z = kx^2 √y

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Textbook Question

In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. x⁴−3x³−20x²−24x−8=0

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Textbook Question

Describe in words the variation shown by the given equation. $z = $\frac{k\sqrt{x}$}{y^2}$

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