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Chapter 9: Problem 64

decide which method of solution applies, and find the solution set. \(|x-4|=1\)

Short Answer

Expert verified
The solution set is \[x = 3, x = 5\].

Step by step solution

01

Identify the Absolute Value Equation

The given equation is \(|x - 4| = 1\). Absolute value equations are of the form \(|A| = B\), where A is an expression and B is a positive number.
02

Set Up Two Separate Equations

For an equation of the form \(|A| = B\), it can be split into two cases: \(A = B\) and \(A = -B\). In this case, set up the two equations: \(x - 4 = 1\) and \(x - 4 = -1\).
03

Solve Each Equation Separately

Solve for \(x\) in each case: \((1) x - 4 = 1 \rightarrow x = 5\) and \((2) x - 4 = -1 \rightarrow x = 3\).
04

Combine the Solutions

The solution set includes both values found: \[x = 3, x = 5\].

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Key Concepts

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

Absolute Value
Understanding the concept of absolute value is crucial when solving absolute value equations. Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is always a non-negative number. For example, the absolute value of both 3 and -3 is 3, because both numbers are 3 units away from zero.

The notation for absolute value is typically written as \(|x|\). So, when we see an equation like \(|x - 4| = 1\), we're dealing with the basic characteristics of absolute value where \(|A| = B \), and we know B must be a positive number.

Key takeaways about absolute value:
  • Absolute value reflects distance, not direction.
  • Absolute value is always positive or zero.
  • \(|-a| = a\) and \(|a| = a\).
Equation Solving
Solving absolute value equations involves understanding how to break the equation into manageable parts. For an equation in the form \(|A| = B\), where B is a positive number, we consider two scenarios: \((1) A = B\) and \((2) A = -B\).

Let's break down the steps applicable to the equation \(|x-4|=1\):
  • First, recognize the absolute value equation: \(|x - 4| = 1\).
  • Next, split this into two separate equations: \(x - 4 = 1\) and \(x - 4 = -1\).
  • Then, solve each equation individually to find the distinct values of x.
Here's how this process looks for our specific example:
1. For \(x - 4 = 1\): Add 4 to both sides to isolate x, i.e., \(x = 5\).
2. For \(x - 4 = -1\): Add 4 to both sides, i.e., \(x = 3\).

By solving these two linear equations, we find the values of x that satisfy the original absolute value equation.
Solution Set
The solution set is the collection of all possible values that satisfy the given equation. In the context of absolute value equations, the solution set will include all values of the variable that make the original equation true.

For the equation \(|x-4|=1\), the steps to determine the solution set were:
  • Splitting \(|x - 4| = 1\) into \(x - 4 = 1\) and \(x - 4 = -1\).
  • Solving each equation to find the values of x.
This gave us two solutions: \(x = 5\) and \(x = 3\).

Hence, the solution set is \(\text{\{3, 5\}}\). This includes all values of x that we derived from solving both cases of the absolute value equation. Remember that you should always combine all solutions found in the separate cases to present a complete solution set.

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Most popular questions from this chapter

For each compound inequality, decide whether intersection or union should be used. Then give the solution set in both interval and graph form. $$ x<5 \text { or } x<-3 $$

For each compound inequality. give the solution set in both interval and graph form. $$ x+1>3 \text { or } x+4<2 $$

Evaluate. $$ -|6|-|-11|+(-4) $$

Decide which method of solution applies, and find the solution set.Graph the solution set. See Examples \(1-3 .\) $$ |5-0.5 x|=4 $$

For each compound inequality, give the solution set in both interval and graph form. $$ x+5 \leq 11 \text { and } x-3 \geq-1 $$
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