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Chapter 8: Problem 32

Solve. $$ |x-3|=8 $$

Short Answer

Expert verified
The solutions are \( x = 11 \) and \( x = -5 \).

Step by step solution

01

- Understand the Absolute Value

The absolute value equation \( |x-3| = 8 \) states that the expression inside the absolute value can be either positive or negative. This means \( x-3 \) can equal 8 or \( x-3 \) can equal -8.
02

- Set Up Two Equations

Set up two separate equations to account for both possible values of the absolute value expression: 1) \( x-3 = 8 \) 2) \( x-3 = -8 \)
03

- Solve the First Equation

Solve the equation \( x-3 = 8 \) for \( x \). Add 3 to both sides: \[ x - 3 + 3 = 8 + 3 \] \[ x = 11 \]
04

- Solve the Second Equation

Solve the equation \( x-3 = -8 \) for \( x \). Add 3 to both sides: \[ x - 3 + 3 = -8 + 3 \] \[ x = -5 \]
05

- State the Solutions

The solutions to the equation \( |x-3| = 8 \) are \( x = 11 \) and \( x = -5 \).

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Key Concepts

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

Understanding Absolute Value
Absolute value refers to the distance a number is from 0 on a number line. No matter whether the number is positive or negative, its absolute value is always non-negative. For instance, the absolute value of both 5 and -5 is 5.
When solving absolute value equations, like \(|x-3|=8\), you need to consider two scenarios:
  • First, the expression inside the absolute value (\(x-3\)) can be equal to 8.
  • Second, the expression inside the absolute value (\(x-3\)) can be equal to -8 because the absolute value of -8 is also 8.
This understanding helps you to set up two separate equations to solve further.
Solving Linear Equations
Linear equations are a type of equation where the highest power of the variable (usually represented by \(x\)) is 1. They are called linear because they represent a straight line when graphed.
In the context of absolute value equations, solving the linear equations is straightforward. Let's take our earlier equations from the absolute value example:
1. \(x - 3 = 8\)
2. \(x - 3 = -8\)
For both, you solve for \(x\) by isolating it on one side of the equation. Here's how:
  • In the first situation, add 3 to both sides to obtain \(x = 11\).
  • In the second situation, add 3 to both sides to get \(x = -5\).
This process is a fundamental skill in algebra, helping you find the variable's value that satisfies the given condition.
Algebraic Solution Methods
When solving absolute value equations, algebraic solution methods involve manipulating the equation algebraically to isolate the variable.
For an equation like \(|x-3|=8\), first, understand the nature of absolute value. Then, set up two separate equations to represent the possible values inside the absolute value.
Next, solve each linear equation step-by-step:
  • Linear equation 1: \(x - 3 = 8\): Add 3 to both sides: \x = 11\.
  • Linear equation 2: \(x - 3 = -8\): Add 3 to both sides: \x = -5\.
Once the solutions are found, combine them to present the final answer: \(x = 11\) and \(x = -5\).
This method ensures that all possible solutions are considered, providing a thorough approach to solving absolute value equations.

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Most popular questions from this chapter

To prepare for Chapter \(9,\) review solving systems of equations using elimination (Section 4.3). Solve. [ 4.3] $$ \begin{aligned} &3 x-5 y=1\\\ &2 x+3 y=7 \end{aligned} $$

For \(g(x)=\frac{2+3 x}{2 x-4},\) find all \(x\) -values for which \(g(x) \geq 0\).

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