Textbook Question
Find the inverse of f(x) = x3 + 2
685
views
Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
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 guidance1
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:
4mWas 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.
Recommended video:
Guided course
3:21
Nonlinear Inequalities
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.
Recommended video:
Guided course
05:46Point-Slope Form
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:
05:18Interval Notation
Related Practice
Textbook Question
Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=3x5+2x4−15x3−10x2+12x+8
489
views
Textbook Question
Find the domain of each function.
589
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
693
views
Textbook Question
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 - 2)/(x + 2) ≤ 2
460
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. h(x) = (x^2 - 3x - 4)/(x^2 - x -6)
573
views