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Chapter 2: Problem 107

Solve. Label any contradictions or identities. $$3|x|-2=10$$

Short Answer

Expert verified
The solutions are \( x = 4 \) and \( x = -4 \).

Step by step solution

01

Isolate the Absolute Value Expression

Start by isolating the absolute value term \(3|x|-2=10\). Add 2 to both sides of the equation to isolate \(3|x|\). \[ 3|x| - 2 + 2 = 10 + 2 \] \[ 3|x| = 12 \]
02

Solve for the Absolute Value

Divide both sides by 3 to solve for \( |x| \). \[ \frac{3|x|}{3} = \frac{12}{3} \] \[ |x| = 4 \]
03

Solve for x

Recognize that \( |x| = 4 \) represents two possible equations: \( x = 4 \) and \( x = -4 \). Therefore, the solutions to the equation are \( x = 4 \) and \( x = -4 \).
04

Check for Contradictions or Identities

Neither \( x = 4 \) nor \( x = -4 \) leads to contradictions. Both satisfy the original equation.

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Key Concepts

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

solving absolute value equations
In mathematics, solving absolute value equations involves dealing with expressions that contain the absolute value function. The absolute value of a number is its distance from zero on the number line, represented as \(|x|\). The key to solving these equations is to recognize that the absolute value \(|x|\) can be either positive or negative. This means that for any equation of the form \(|x| = a\), where \(a\) is a positive number, there are two possible solutions: \(x\) can be \(a\) or \(-a\).
For example, if we have \(|x| = 4\), the solutions will be \(x = 4\) and \(x = -4\). Always remember to consider both possibilities.
isolating absolute value
When solving absolute value equations, the first step usually involves isolating the absolute value expression. Let's look at the equation from the exercise: \(3|x| - 2 = 10\).
  • We start by adding 2 to both sides to eliminate the constant term outside the absolute value expression:
\[3|x| - 2 + 2 = 10 + 2\]
This simplifies to:
\[3|x| = 12\]
  • Next, we divide both sides by 3 to completely isolate \(|x|\):
\[\frac{3|x|}{3} = \frac{12}{3}\]
\[|x| = 4\]
Now that we have isolated \(|x|\), we can proceed to solving for \(x\).
checking for contradictions
After solving an absolute value equation, it is important to check for any contradictions. Contradictions occur when substituting the solutions back into the original equation yields false statements.
For the exercise provided, once we isolated \(|x| = 4\), we recognized that \(x\) could be 4 or -4. To check for contradictions, we substitute these solutions back into the original equation: \(3|x| - 2 = 10\).
  • For \(x = 4\):
\[3|4|-2 = 12 - 2 = 10\] which is true.
  • For \(x = -4\):
\[3|-4|-2 = 12 - 2 = 10\] which is also true.
Both solutions satisfy the original equation, confirming there are no contradictions. This step ensures the solutions are correct.

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