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 51

Chapter 4, Problem 51

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

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Identify the critical points by setting the numerator and denominator equal to zero separately. For the numerator: \(x = 0\). For the denominator: \(x - 3 = 0\), so \(x = 3\). These points divide the number line into intervals to test.

Determine the intervals created by the critical points: \((-\infty, 0)\), \((0, 3)\), and \((3, \infty)\). These intervals will be tested to see where the inequality \(\frac{x}{x-3} > 0\) holds true.

Choose a test point from each interval and substitute it into the expression \(\frac{x}{x-3}\). Check the sign (positive or negative) of the result to determine if the inequality is satisfied in that interval.

Remember that the inequality is strict (\(>\) 0), so exclude points where the expression is zero or undefined. Specifically, exclude \(x = 0\) (where numerator is zero) and \(x = 3\) (where denominator is zero and expression is undefined).

Combine the intervals where the expression is positive to write the solution set in interval notation, and then graph these intervals on the real number line, marking excluded points appropriately.

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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 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.

Critical Points and Sign Analysis

Critical points occur where the numerator or denominator equals zero, dividing the number line into intervals. By testing values in each interval, you determine the sign of the rational expression, which helps identify where the inequality holds true.

Interval Notation and Graphing Solutions

Interval notation concisely represents sets of real numbers that satisfy the inequality, using parentheses or brackets to indicate whether endpoints are included. Graphing on a number line visually shows these solution intervals and excluded points, aiding interpretation.

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Interval 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)=−x³+x²+16x−16

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Textbook Question

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. 2x⁵+7x⁴−18x²−8x+8=0

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Textbook Question

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.

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Textbook Question

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

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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+4)/x>0

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Textbook Question

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)

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