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 60

Chapter 4, Problem 60

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/(x + 2) ≥ 2

Verified step by step guidance

Start by rewriting the inequality: $\frac{x}{x + 2} \geq 2$.

Bring all terms to one side to have zero on the other side: $\frac{x}{x + 2} - 2 \geq 0$.

Find a common denominator and combine the terms: $\frac{x - 2(x + 2)}{x + 2} \geq 0$.

Simplify the numerator: $\frac{x - 2x - 4}{x + 2} = \frac{-x - 4}{x + 2} \geq 0$.

Determine the critical points by setting numerator and denominator equal to zero: numerator $-x - 4 = 0$ and denominator $x + 2 = 0$. These points divide the number line into intervals to test for the inequality.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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 for the inequality.

Interval Notation and Graphing on a Number Line

Interval notation concisely represents sets of real numbers that satisfy the inequality, using parentheses or brackets to indicate open or closed intervals. Graphing these intervals on a number line visually shows the solution set and helps verify the correctness of the solution.

Recommended video:

05:18

Interval Notation

Related Practice

Textbook Question

Find the inverse of f(x) = x³ + 2

Textbook Question

Solve: $$\sqrt{x}$ + $\sqrt{x + 5}$ = 5$

views

Textbook Question

Find the domain of each function. $f(x) = $\sqrt{2x^2 - 5x + 2}$$

views

Textbook Question

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. r(x) = (x^2 + 4x + 3)/(x + 2)^2

Textbook Question

Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product?

Textbook Question

Follow the seven steps to graph each rational function. f(x)=2x/(x²−4)