Explanation Procedure used: (1) . Find an equivalent inequality with 0 on one side. (2). Change the inequality symbol to an equals sign and solve the related equation. (3). Find values of the variables for which the related rational function is not defined. (4). The numbers found in steps (2) and (3) are called the critical values. Use the critical values to divide the x - axis into intervals. Then determine the function’s sign in each interval using an x value from the interval or using the graph of the equation. (5) Select the intervals for which the inequality is satisfied and write intervals notation or set builder notation for the solution set. If the inequality symbol ≤ or ≥ , then the solutions to step (2) should be include in the solution set. The x values found in step (3) are never included in the solution set. Calculation: The given polynomial inequality is h ( x ) ≥ 0 , where h ( x ) = 7 x ( x − 1 ) ( x + 5 ) . By the above procedure, firstly obtain the solution set of the inequality. That is, obtain the solution of the equation 7 x ( x − 1 ) ( x + 5 ) = 0 . The rational equation is undefined at the points x = − 5 and x = 1 . Obtain the critical point of the equation 7 x ( x − 1 ) ( x + 5 ) = 0 by multiplying ( x − 1 ) ( x + 5 ) on both sides. 7 x ( x − 1 ) ( x + 5 ) = 0 ( x − 1 ) ( x + 5 ) ⋅ ( 7 x ( x − 1 ) ( x + 5 ) ) = 0 ⋅ ( x − 1 ) ( x + 5 ) 7 x = 0 x = 0 Thus, the critical values of the rational equation are 0 , − 5 , and 1 . The critical values divide the x axis into the intervals as ( − ∞ , − 5 ) , ( − 5 , 0 ) , ( 0 , 1 ) , and ( 1 , ∞ ) . Find the sign of the rational function on each interval by selecting test value as follows. Obtain the sign of polynomial h ( x ) = 7 x ( x − 1 ) ( x + 5 ) at x = − 6 on the interval ( − ∞ , − 5 ) . h ( − 6 ) = 7 ( − 6 ) ( − 6 − 1 ) ( − 6 + 5 ) = − 42 ( − 7 ) ( − 1 ) = − 6 < 0 Obtain the sign of polynomial h ( x ) = 7 x ( x − 1 ) ( x + 5 ) at x = 1 on the interval ( − 5 , 0 )

College Algebra: Graphs and Models (6th Edition)

6th Edition

ISBN: 9780134179032

Author: Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen, Judith A. Penna

Publisher: PEARSON

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 4.6, Problem 13E

To determine

To solve: The rational inequality h(x)≥0, where h(x)=7x(x−1)(x+5).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.6, Problem 12E

Chapter 4.6, Problem 14E

Students have asked these similar questions

G=4/5 - 1/3 x 2/15 + 4/45

f(x) = 5x2 + 1 when x = 2.9999

Which value, to the nearest tenth, is the smallest solution of f(x) = g(x) if f\x) = In(x+ 5) - 1 and g(x)= x³ - 2x+ 1? -3.2 O 1.6 2.3 -1.6

Chapter 4 Solutions

College Algebra: Graphs and Models (6th Edition)

Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 2E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 10E

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

The rational inequality h ( x ) ≥ 0 , where h ( x ) = 7 x ( x − 1 ) ( x + 5 ) .

Solution Summary: The author explains how to solve the rational inequality h(x)ge 0.

Concept explainers

Videos

To solve: The rational inequality h(x)≥0, where h(x)=7x(x−1)(x+5).

Want to see the full answer?

Chapter 4 Solutions