Skip to main content
Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 52
Capitolo 4, Problema 52

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)/x>0

Guida verificata passo dopo passo
1
Identify the critical points by setting the numerator and denominator equal to zero separately: solve \( x + 4 = 0 \) and \( x = 0 \). These points divide the number line into intervals.
The critical points are \( x = -4 \) and \( x = 0 \). Use these points to split the real number line into three intervals: \( (-\infty, -4) \), \( (-4, 0) \), and \( (0, \infty) \).
Determine the sign of the rational expression \( \frac{x+4}{x} \) on each interval by choosing a test point from each interval and substituting it into the expression.
Analyze the inequality \( \frac{x+4}{x} > 0 \) by selecting intervals where the expression is positive based on the sign test results.
Express the solution set in interval notation, remembering to exclude points where the expression is undefined (denominator zero) and to consider whether to include points where the numerator is zero based on the strict inequality.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
5m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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 finding where the expression is positive, negative, or zero by analyzing the signs of numerator and denominator.
Video consigliato:
3:21
Nonlinear Inequalities

Critical Points and Sign Analysis

Critical points are values that make the numerator or denominator 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.
Video consigliato:
05:46
Point-Slope Form

Interval Notation and Graphing Solutions

Interval notation concisely represents sets of real numbers that satisfy inequalities, using parentheses or brackets to indicate open or closed intervals. Graphing on a number line visually shows these solution sets, highlighting excluded points like zeros of the denominator.
Video consigliato:
05:18
Interval Notation
Pratica correlata
Domanda del libro di testo

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)=−x3+x2+16x−16

530
views
Domanda del libro di testo

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)(x−1)/(x+2)≤0

475
views
Domanda del libro di testo

In Exercises 51–54, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1.

704
views
Domanda del libro di testo

Use transformations of f(x)=1/x or f(x)=1/x2 to graph each rational function. h(x)=1/x2 − 4

861
views
Domanda del libro di testo

Write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 2x2 but with the given point as the vertex. (−10, −5)

1031
views
Domanda del libro di testo

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−3)>0

557
views