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Chapter 3: Problem 1

Solve. $$|x|=7$$

Short Answer

Expert verified
x = 7 and x = -7

Step by step solution

01

Understand the Definition of Absolute Value

Absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, o matter whether the number is positive or negative, its absolute value is always non-negative.
02

Set Up Equations

Given o \begin assign two cases to the absolute value equation: \(|x|=7\) can be translated into:1. \begin x = 7\begin2. \begin x = -7
03

Solve the Equations

Solve each of the equations from Step 2. For \(x = 7\), the solution is simply \begin x = 7.For \(x = -7\), the solution is \begin x = -7.Hence, the solutions to the equation are \begin x = 7\begin and \begin x = -7.

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Key Concepts

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

Absolute Value
The absolute value of a number measures its distance from zero on the number line, without considering the direction. It is always a non-negative number. For example, the absolute value of both 7 and -7 is 7. This is because both are 7 units away from zero. The absolute value is denoted by vertical bars: \(|x|\). So, \(|7| = 7\) and \(|-7| = 7\). This concept is vital when solving absolute value equations because it helps us understand why we have two potential solutions for each equation.
Equation Solving
When solving equations involving absolute values, we derive two possible equations. Given \(|x| = 7\), we translate this into two simpler equations: one for the positive scenario and one for the negative.

Here鈥檚 the process:
  • Consider \(x = 7\)
  • Consider \(x = -7\) as the other possibility
We set up these two equations because \(|x| = 7\) means that the distance of \(x\) from zero is 7 units, which can be in either positive or negative direction. Solving these simple equations gives us two solutions, \(x = 7\) and \(x = -7\).
Number Line
Visualizing absolute value equations using a number line can make understanding easier. The number line represents numbers as points along a line, with zero in the center and positive numbers to the right and negative numbers to the left.

Let's apply this to our problem, \(|x|=7\):
  • Locate 0 on the number line
  • Mark 7 units to the right of 0 (which is +7)
  • Mark 7 units to the left of 0 (which is -7)
This visualization helps us see why there are two solutions: \(x = 7\) and \(x = -7\), because both points are the same distance from zero, reflecting the absolute value principle.

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