Question

johan found that the equation $-2|8 - x| - 6 = -12$ had two possible solutions: $x = 5$ and $x = -11$. which explains whether his solutions are correct?

he is not correct because there are no solutions.

he is not correct because he made a sign error.

he is correct because both solutions satisfy the equation.

he is not correct because there is only one solution: $x = 5$.

Explanation:

Step1: Solve the equation \(-2|8 - x| - 6 = -12\)

First, isolate the absolute value term. Add 6 to both sides:
\(-2|8 - x| = -12 + 6\)
\(-2|8 - x| = -6\)
Then divide both sides by -2:
\(|8 - x| = \frac{-6}{-2}\)
\(|8 - x| = 3\)

Step2: Solve the absolute value equation \(|8 - x| = 3\)

This gives two cases:
Case 1: \(8 - x = 3\)
Subtract 8 from both sides:
\(-x = 3 - 8\)
\(-x = -5\)
Multiply both sides by -1:
\(x = 5\)

Case 2: \(8 - x = -3\)
Subtract 8 from both sides:
\(-x = -3 - 8\)
\(-x = -11\)
Multiply both sides by -1:
\(x = 11\) (not -11, so Johan made a sign error here)

Answer:

He is not correct because he made a sign error.