Certainly! Let's solve the equation [tex]\(|6x - 6| = 30\)[/tex] step by step.
## Step-by-Step Solution
### Step 1: Understanding Absolute Values
The absolute value equation [tex]\(|A| = B\)[/tex] has two possible cases:
1. [tex]\(A = B\)[/tex]
2. [tex]\(A = -B\)[/tex]
Given [tex]\(|6x - 6| = 30\)[/tex], we can set up two separate equations to solve for [tex]\(x\)[/tex].
### Step 2: Set Up the Equations
From the absolute value equation, we derive the two cases:
1. [tex]\(6x - 6 = 30\)[/tex]
2. [tex]\(6x - 6 = -30\)[/tex]
### Step 3: Solve the First Case
Solve the equation [tex]\(6x - 6 = 30\)[/tex]:
1. Add 6 to both sides to isolate the term containing [tex]\(x\)[/tex]:
[tex]\[ 6x - 6 + 6 = 30 + 6 \][/tex]
[tex]\[ 6x = 36 \][/tex]
2. Divide both sides by 6 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{36}{6} \][/tex]
[tex]\[ x = 6 \][/tex]
### Step 4: Solve the Second Case
Solve the equation [tex]\(6x - 6 = -30\)[/tex]:
1. Add 6 to both sides to isolate the term containing [tex]\(x\)[/tex]:
[tex]\[ 6x - 6 + 6 = -30 + 6 \][/tex]
[tex]\[ 6x = -24 \][/tex]
2. Divide both sides by 6 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{-24}{6} \][/tex]
[tex]\[ x = -4 \][/tex]
### Step 5: Summary of Solutions
The solutions to the equation [tex]\(|6x - 6| = 30\)[/tex] are:
[tex]\[ x = 6 \quad \text{and} \quad x = -4 \][/tex]
Thus, the values of [tex]\(x\)[/tex] that satisfy the given equation are:
[tex]\[ x = 6.0 \quad \text{and} \quad x = -4.0 \][/tex]
