Questions: Solve the following absolute value equation. If needed, write your answer as a fraction reduced to lowest terms. 6z-6=9

Solve the following absolute value equation. If needed, write your answer as a fraction reduced to lowest terms. 6z-6=9
Transcript text: Solve the following absolute value equation. If needed, write your answer as a fraction reduced to lowest terms. \[ |6 z-6|=9 \]
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Solution

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Solution Steps

To solve the absolute value equation \(|6z - 6| = 9\), we need to consider the definition of absolute value. The equation \(|A| = B\) implies two possible equations: \(A = B\) and \(A = -B\). Therefore, we will split the given equation into two separate linear equations and solve each for \(z\).

Solution Approach
  1. Split the absolute value equation into two separate equations:
    • \(6z - 6 = 9\)
    • \(6z - 6 = -9\)
  2. Solve each equation for \(z\).
  3. Check if the solutions are valid.
Step 1: Split the Absolute Value Equation

Given the equation \(|6z - 6| = 9\), we split it into two separate linear equations:

  1. \(6z - 6 = 9\)
  2. \(6z - 6 = -9\)
Step 2: Solve Each Equation

Solve the first equation: \[ 6z - 6 = 9 \] Add 6 to both sides: \[ 6z = 15 \] Divide by 6: \[ z = \frac{15}{6} = \frac{5}{2} \]

Solve the second equation: \[ 6z - 6 = -9 \] Add 6 to both sides: \[ 6z = -3 \] Divide by 6: \[ z = \frac{-3}{6} = \frac{-1}{2} \]

Final Answer

The solutions to the equation \(|6z - 6| = 9\) are: \[ \boxed{z = \frac{5}{2} \text{ or } z = \frac{-1}{2}} \]

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