/*! 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 57 Solve each absolute value equati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 57

Solve each absolute value equation. $$\left|x^{2}-x-6\right|=6$$

Short Answer

Expert verified
x = 4, x = -3, x = 0, x = 1

Step by step solution

01

- Understand the Absolute Value Equation

An absolute value equation \(\right|A\right|=B\) means that the expression inside the absolute value can be equal to \(B\) or \(-B\). Here, \(\right|x^{2}-x-6\right|=6\) implies \(x^2 - x - 6 = 6\) or \(x^2 - x - 6 = -6\).
02

- Set Up and Solve the First Equation

Solve \(x^2 - x - 6 = 6\): \[ x^2 - x - 6 = 6 \] \[ x^2 - x - 12 = 0 \] Factorize: \[ (x - 4)(x + 3) = 0 \] Thus, \[ x = 4 \ or \ x = -3 \].
03

- Set Up and Solve the Second Equation

Solve \(x^2 - x - 6 = -6\): \[ x^2 - x - 6 = -6 \] \[ x^2 - x = 0 \] Factorize: \[ x(x - 1) = 0 \] Thus, \[ x = 0 \ or \ x = 1 \].
04

- List All Solutions

Combine all the solutions from both equations: \[ x = 4, x = -3, x = 0, x = 1 \]

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving absolute value equations
Absolute value equations might seem tricky, but they can be simplified with a systematic approach. An absolute value equation \(\left|A\right|=B\) indicates that what's inside the absolute value (A) could be equal to B or -B. Understanding this concept is crucial as it lays the foundation for solving the equation.
For instance, consider the equation \(\left|x^{2}-x-6\right|=6\). This implies two separate equations: \(x^2 - x - 6 = 6\) and \(x^2 - x - 6 = -6\).
This understanding allows us to methodically break down the problem and solve it using standard algebraic methods, such as equations and factoring.
factoring quadratic equations
Factoring quadratic equations is a key skill in algebra, especially when solving absolute value equations. When given \(x^2 - x - 6 = 6\), we first rearrange the equation to standard form: \(x^2 - x - 12 = 0\). To factor, we look for two numbers that multiply to -12 and add to -1, which are -4 and 3. Thus, we factorize as so: \((x - 4)(x + 3) = 0\).
This results in two solutions: \(x = 4\) and \(x = -3\).
Dealing with the second equation, \(x^2 - x - 6 = -6\), we simplify it to \(x^2 - x = 0\), factoring it as \(x(x - 1) = 0\), yielding \(x = 0\) and \(x = 1\).
Mastering factoring is crucial for solving such algebraic problems with ease.
algebraic solutions
Combining algebraic solutions is a final step after solving individual equations. For our absolute value equation \(\left|x^{2}-x-6\right|=6\), we combined results from both sets of equations.
From \(x^2 - x - 12 = 0\), we got \(x = 4\) and \(x = -3\).
From \(x^2 - x = 0\), the solutions were \(x = 0\) and \(x = 1\).
Together, these yield \(x = 4, x = -3, x = 0, \) and \(x = 1\).
This step emphasizes the importance of understanding and applying algebraic methods uniformly to achieve a comprehensive set of solutions.

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

If \(y\) varies inversely as \(x\) and \(y=3\) when \(x=12,\) find \(y\) when \(x=40\)

Solve each polynomial inequality using the test-point method. $$x^{4}-5 x^{3}+3 x^{2}+15 x-18 \geq 0$$

Use division to write each rational expression in the form quotient \(+\) remainder/divisor. Use synthetic division when possible. $$\frac{4 t-5}{2 t+1}$$

Write the function \(f(x)=2 x^{2}-3 x+1\) in the form \(f(x)=a(x-h)^{2}+k\)

Solve the problem. Admission to the Zoo Winona paid 100 dollar for a lifetime membership to Friends of the Zoo, so that she could gain admittance to the zoo for only 1 dollar per visit. Write Winona's average cost per visit \(C\) as a function of the number of visits when she has visited \(x\) times. What is her average cost per visit when she has visited the zoo 100 times? Graph the function for \(x>0 .\) What happens to her average cost per visit if she starts when she is young and visits the zoo every day?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

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.