Skip to main content
Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 3 - Polynomial and Rational Functions문제 57
4장, 문제 57

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 + 4)/(2x - 1) ≤ 3

검증된 단계별 안내
1
Start by rewriting the inequality to have zero on one side: subtract 3 from both sides to get \(\frac{\!x + 4}{2x - 1} - 3 \leq 0\).
Find a common denominator and combine the terms into a single rational expression: \(\frac{\!x + 4}{2x - 1} - \frac{3(2x - 1)}{2x - 1} \leq 0\), which simplifies to \(\frac{\!x + 4 - 3(2x - 1)}{2x - 1} \leq 0\).
Simplify the numerator by distributing and combining like terms: \(x + 4 - 6x + 3 = -5x + 7\), so the inequality becomes \(\frac{-5x + 7}{2x - 1} \leq 0\).
Identify critical points by setting numerator and denominator equal to zero separately: solve \(-5x + 7 = 0\) and \(2x - 1 = 0\) to find values where the expression changes sign or is undefined.
Use the critical points to divide the number line into intervals, then test a value from each interval in the inequality \(\frac{-5x + 7}{2x - 1} \leq 0\) to determine where the inequality holds true. Finally, express the solution set in interval notation and graph it on the real number line.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
4m
도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to a number or another expression. Solving them requires finding values of the variable that make the inequality true, considering where the denominator is zero to avoid undefined expressions.
추천 영상:
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 the sign of the rational expression, which helps identify where the inequality holds true.
추천 영상:
05:46
Point-Slope Form

Interval Notation and Graphing Solutions

Interval notation expresses the solution set compactly using parentheses and brackets to indicate open or closed intervals. Graphing on a number line visually represents these intervals, showing where the inequality is satisfied and highlighting excluded points like zeros of the denominator.
추천 영상:
05:18
Interval Notation
관련 실천
교과서 질문

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

596
views
교과서 질문

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 + 1)/(x + 3) < 2

520
views
교과서 질문

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. f(x) = 2x/(x^2 - 9)

702
views
교과서 질문

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)=2x4−3x3−7x2−8x+6

427
views
교과서 질문

Use transformations of f(x) = (1/x) or f(x) = (1/x2) to graph each rational function. g(x) = 1/(x + 2)2 - 1

1039
views
교과서 질문

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.1/(x - 3) < 1

535
views