To solve the equation [tex]\(|6x - 4| = 8\)[/tex], we must consider the nature of absolute value. The absolute value equation [tex]\( |A| = B \)[/tex] implies that [tex]\( A = B \)[/tex] or [tex]\( A = -B \)[/tex]. Applying this to our problem, we set up two separate equations:
### Case 1: [tex]\( 6x - 4 = 8 \)[/tex]
1. Solve for [tex]\( x \)[/tex]:
[tex]\[ 6x - 4 = 8 \][/tex]
2. Add 4 to both sides:
[tex]\[ 6x = 12 \][/tex]
3. Divide both sides by 6:
[tex]\[ x = \frac{12}{6} \][/tex]
4. Simplify the division:
[tex]\[ x = 2 \][/tex]
### Case 2: [tex]\( 6x - 4 = -8 \)[/tex]
1. Solve for [tex]\( x \)[/tex]:
[tex]\[ 6x - 4 = -8 \][/tex]
2. Add 4 to both sides:
[tex]\[ 6x = -4 \][/tex]
3. Divide both sides by 6:
[tex]\[ x = \frac{-4}{6} \][/tex]
4. Simplify the division:
[tex]\[ x = -\frac{2}{3} \][/tex]
Therefore, the solutions to the equation [tex]\(|6x - 4| = 8\)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = -\frac{2}{3} \)[/tex]. In comma-separated form, the answers are:
[tex]\[ x = 2, -\frac{2}{3} \][/tex]
