/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 66 Solve each inequality and graph ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 66

Solve each inequality and graph the solution set on a real number line. $$\left|x^{2}+6 x+1\right|>8$$

Short Answer

Expert verified
The solution to the inequality \(\left|x^{2}+6 x+1\right|>8\) is \(x < -7\) or \(x > 1\).

Step by step solution

01

Express the inequality without the absolute value

Begin by expressing the inequality without the absolute value involved. Since the absolute value of a quantity is either positive or zero, we write the inequality as two separate cases: \(x^{2}+6x+1 > 8\) and \(x^{2}+6x+1 < -8\). Both of these inequalities must be solved individually.
02

Solving first inequality

We start by solving the inequality \(x^{2}+6x+1 > 8\). To make it easier, rearrange the equation to bring all terms to one side. It simplifies to \(x^{2}+6x-7 > 0\). Now, solve the inequality by factoring it to \((x+7)(x-1) > 0\). Using the sign table, we can see that the solution set for this inequality is \(x < -7\) or \(x > 1\).
03

Solving second inequality

Now we will solve the second inequality \(x^{2}+6x+1 < -8\). Rearranging it, we get \(x^{2}+6x+9 < 0\). However, since the left side of the equation is a perfect square (it can be expressed as \((x+3)^{2}\)), it can never be negative, indicating that this inequality has no solution.
04

Graphing the solution on a number line

The given inequality solution can be represented on a number line. The first inequality's solutions will be graphed as open circles at \(x = -7\) and \(x = 1\), with \(x < -7\) graphed to left of -7 and \(x > 1\) graphed to the right of 1. And second inequality does not provide any solution. Hence, there will be no graph for this.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When all is said and done, it seems to me that direct variation equations are special kinds of linear functions and inverse variation equations are special kinds of rational functions.

Basic Car Rental charges \(\$ 20\) a day plus \(\$ 0.10\) per mile, whereas Acme Car Rental charges \(\$ 30\) a day plus \(\$ 0.05\) per mile. How many miles must be driven to make the daily cost of a Basic Rental a better deal than an Acme Rental?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Write a rational inequality whose solution set is \((-\infty,-4) \cup[3, \infty)\).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When solving \(f(x)>0,\) where \(f\) is a polynomial function, I only pay attention to the sign of \(f\) at each test value and not the actual function value.

A company that manufactures running shoes has a fixed monthly cost of \(\$ 300,000\). It costs \(\$ 30\) to produce each pair of shoes. a. Write the cost function, \(C,\) of producing \(x\) pairs of shoes. b. Write the average cost function, \(\bar{C},\) of producing \(x\) pairs of shoes. c. Find and interpret \(\bar{C}(1000), \bar{C}(10,000),\) and \(\bar{C}(100,000)\) d. What is the horizontal asymptote for the graph of the average cost function, \(\bar{C} ?\) Describe what this represents for the company.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.