Question

what are the solutions of the following equation?
$4|x - 8| + 7 = -5|x - 8| - 29$
choose 1 answer:
a $x = 4$ or $x = 12$
b $x = \frac{1}{2}$ or $x = -\frac{1}{2}$
c $x = -4$ or $x = -12$
d only $x = -4$
e there are no solutions

Explanation:

Step1: Let \( y = |x - 8| \)

Substitute \( y \) into the equation: \( 4y + 7 = -5y - 29 \)

Step2: Solve for \( y \)

Add \( 5y \) to both sides: \( 4y + 5y + 7 = -29 \)
Simplify: \( 9y + 7 = -29 \)
Subtract 7 from both sides: \( 9y = -29 - 7 \)
Calculate: \( 9y = -36 \)
Divide by 9: \( y = \frac{-36}{9} = -4 \)

Step3: Analyze \( y = |x - 8| \)

The absolute value \( |x - 8| \) is always non - negative (i.e., \( |x - 8| \geq 0 \) for all real \( x \)). But we found \( y=-4 \), and \( - 4<0 \), which is a contradiction. So there are no solutions for \( x \) that satisfy the original equation.

Answer:

There are no solutions (corresponding to the option: There are no solutions)