/*! 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 44 \(|3 x-1|=-4\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 44

\(|3 x-1|=-4\)

Short Answer

Expert verified
No solution exists because an absolute value can't be negative.

Step by step solution

01

Understand Absolute Value

The absolute value of a number, expressed as |x|, is always non-negative. It represents the distance of a number from zero on a number line without considering direction. So, |x| must always be greater than or equal to zero.
02

Analyze the Problem Equation

Given the equation \(|3x - 1| = -4\). Recognize that an absolute value cannot equal a negative number as discussed in Step 1. This immediately tells us that there is no solution because it's impossible for any expression within absolute value to result in a negative number.

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.

Negative Numbers and Absolute Values
Negative numbers are values less than zero, represented with a minus sign (-). They are the opposite of positive numbers and appear on the left side of zero on the number line.

It's crucial to know that absolute values make negative numbers positive. Absolute value, given by the symbol \(|x|\), indicates how far a number is from zero, without considering if it's to the left or right on the number line.

For example:
  • |-3| equals 3 because -3 is three units away from zero.
  • |5| remains 5, since it's also five units from zero, but to the right.
Therefore, inside an absolute value, all numbers are treated as non-negative.
Solving Equations with Absolute Values
Solving equations involves finding the value of unknown variables that make the equation true. In equations with absolute values, the goal is to account for all possible distances from zero.

Consider an equation like \(|x| = a\). If 'a' is a positive number, the equation \(x = a\) and \(x = -a\) are both possible solutions because both values are equally distanced from zero. However, what happens when 'a' is negative?

In such cases, there are no solutions for \(|x| = a\), as absolute values cannot be negative. This is due to their intrinsic nature that expresses non-negative distances.
Recognizing When There Is No Solution
In some mathematical problems, it is key to recognize scenarios where solutions can't exist.

The equation \(|3x - 1| = -4\) is one such instance. Since absolute values represent non-negative outcomes, having an equation equate to a negative number means there's a fundamental conflict.

No matter how you try to solve it, no values for \(x\) will satisfy this statement because absolute values cannot be negative. Understanding and accepting that some equations have no solutions is an important skill in mathematics.

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

For Problems \(1-12\), write each radical in terms of \(i\) and simplify; for example, \(\sqrt{-20}=i \sqrt{20}=i \sqrt{4} \cdot \sqrt{5}=\) \(2 i \sqrt{5}\) $$ \sqrt{-54} $$

For Problems \(35-54\), find each product and express it in the standard form of a complex number \((a+b i)\). $$ -4 i(-5-6 i) $$

For Problems \(13-34\), add or subtract the complex numbers as indicated. $$ (4-i)+(4+i) $$

Solve each of the following quadratic equations, and check your solutions. $$7 x^{2}+3 x+3=0$$

For Problems \(35-54\), find each product and express it in the standard form of a complex number \((a+b i)\). $$ (7 i)(8 i) $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

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.