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Chapter 11: Problem 54

Solve. $$ 4|x|=48 $$

Short Answer

Expert verified
The solutions are \( x = 12 \) and \( x = -12 \).

Step by step solution

01

Isolate the Absolute Value

To start solving, divide both sides of the equation by 4 to isolate the absolute value: \[ 4|x| = 48 \] Divide by 4: \[ |x| = \frac{48}{4} \] \[ |x| = 12 \]
02

Consider the Definition of Absolute Value

The definition of absolute value tells us that if \(|x| = 12\), then there are two possible values for \(x\): \[ x = 12 \] and \[ x = -12 \]
03

Write the Solution

Combine the two possible values to express the solution set: \[ x = 12 \text{ or } x = -12 \]

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Key Concepts

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

isolate absolute value
When solving equations involving absolute values, the first step is to isolate the absolute value expression. Here, isolate means to get the absolute value expression by itself on one side of the equation.
In the given problem, we have:
4|x| = 48
We accomplish isolation by dividing both sides by 4:
\[ 4|x| / 4 = 48 / 4 \]
Simplifying this gives us:
\[|x| = 12 \]
Once the absolute value expression is isolated, we can move to the next step.
absolute value definition
Absolute value is a concept in mathematics that represents the distance of a number from zero on the number line, without considering direction. The absolute value of a number is always non-negative.
The absolute value of a number \(x\) is denoted as \(|x|\). By definition:
  • If \(x \gt 0\), then \(|x| = x\).
  • If \(x = 0\), then \(|x| = 0\).
  • If \(x \lt 0\), then \(|x| = -x\).

In our problem, once we isolated the absolute value expression:
\[|x| = 12 \]
This means the distance of \(x\) from 0 is 12. Considering the definition, \(x\) can be 12 or -12.
solution set
The solution set is the collection of values that satisfy the given equation. To determine the solution set for absolute value equations, we consider both possible cases that were identified using the absolute value definition.
For \(|x| = 12\):
  • First possibility: \(x = 12\).
  • Second possibility: \(x = -12\).

Combining these results, we write the solution set:
\[ x = 12 \text{ or } x = -12 \]
This tells us that both 12 and -12 satisfy the original equation.

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