/*! 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 78 Solve each equation or inequalit... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 78

Solve each equation or inequality. Graph the solution set. $$ |5-0.5 x|=4 $$

Short Answer

Expert verified
The solutions are x = 2 and x = 18.

Step by step solution

01

Understand the Absolute Value Equation

An absolute value equation |A| = B can be split into two separate equations: A = B and A = -B. Here, we need to solve for the variable x using these properties. So we have two equations: 5 - 0.5x = 4 and 5 - 0.5x = -4.
02

Solve the First Equation

For the first equation: 5 - 0.5x = 4. Subtract 5 from both sides to get: -0.5x = -1. Then divide by -0.5: x = 2.
03

Solve the Second Equation

For the second equation: 5 - 0.5x = -4. Subtract 5 from both sides to get: -0.5x = -9. Then divide by -0.5: x = 18.
04

Graph the Solution Set

The solutions of the equation are x = 2 and x = 18. On a number line, place points at x = 2 and x = 18 to represent the solution set.

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 equations
Solving equations means finding the value(s) of the variable that make the equation true. In our problem, we are solving the absolute value equation \(|5 - 0.5x| = 4\). The first step is to understand that an absolute value equation \(|A| = B\) can be split into two cases: \(A = B\) and \(A = -B\). This method helps us handle the absolute value and solve for the variable inside it.
Graphing solutions
Once we solve the equations, we need to graph the solutions to visualize them. For our equations, the solutions are \(x = 2\) and \(x = 18\). To graph these solutions, we use a number line. A number line is a horizontal line where each point represents a real number. Place a dot at \(x = 2\) and another dot at \(x = 18\) on the line. These points represent the solutions to the absolute value equation.
Algebraic manipulation
Algebraic manipulation involves using algebraic methods to rearrange and solve equations. For the first equation \((5 - 0.5x = 4)\), we subtract 5 from both sides to get \(-0.5x = -1\). Then, we divide by -0.5 to solve for x, resulting in \( x = 2 \). For the second equation \((5 - 0.5x = -4)\), we again subtract 5 from both sides, resulting in \(-0.5x = -9\), and dividing by -0.5 gives us \( x = 18 \). Both solutions must be checked back into the original absolute value equation to ensure they鈥檙e correct.
Absolute value properties
Absolute value represents the distance of a number from zero on a number line, regardless of direction. Key properties of absolute values are:
  • \(|A| \geq 0 \): Absolute value is always non-negative.
  • \(| A | = | -A | \): Both a number and its negative have the same absolute value.
  • An absolute value equation \( | A | = B \) can be written as \( A = B \) and \( A = - B \).
Knowing these properties helps us understand and solve equations involving absolute values efficiently.

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

Solve each equation or inequality. $$ |7 x+4|=0 $$

After 1 yr on the job, Grady got a raise from \(\$ 10.50\) per hour to \(\$ 11.76\) per hour. What was the percent increase in his hourly wage?

Solve each problem. How many liters of a \(14 \%\) alcohol solution must be mixed with \(20 \mathrm{~L}\) of a \(50 \%\) solution to obtain a \(30 \%\) solution?

Speedy Delivery finds that the cost of making \(x\) deliveries is \(C=3 x+2300,\) while the revenue produced from them is \(R=5.50 x(C\) and \(R\) in dollars).

Give, in interval notation, the unknown numbers in each description. When 1 is added to twice a number, the result is greater than or equal to \(7 .\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

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.