Question

johan found that the equation $-2|8 - x| - 6 = -12$ had two possible solutions: $x = 5$ and $x = -1$ whether his solutions are correct?

• he is correct because both solutions satisfy the equation.
• he is not correct because he made a sign error.
• he is not correct because there are no solutions.
• he is not correct because there is only one solution: $x = 5$.

Explanation:

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

First, add 6 to both sides of the equation:
\(-2|8 - x| - 6 + 6 = -12 + 6\)
\(-2|8 - x| = -6\)

Step2: Divide both sides by -2

\(\frac{-2|8 - x|}{-2} = \frac{-6}{-2}\)
\(|8 - x| = 3\)

Step3: Solve the absolute - value equation

The absolute - value equation \(|A|=b\) (where \(b\geq0\)) has two solutions: \(A = b\) or \(A=-b\).
For \(|8 - x| = 3\), we have two cases:

Case 1: \(8 - x=3\)

Subtract 8 from both sides:
\(8 - x-8=3 - 8\)
\(-x=-5\)
Multiply both sides by - 1: \(x = 5\)

Case 2: \(8 - x=-3\)

Subtract 8 from both sides:
\(8 - x-8=-3 - 8\)
\(-x=-11\)
Multiply both sides by - 1: \(x = 11\)

Now we check the solutions Johan provided (\(x = 5\) and \(x=-1\)):

• For \(x = 5\):

Substitute \(x = 5\) into the left - hand side of the original equation \(-2|8 - x|-6\):
\(-2|8 - 5|-6=-2\times|3|-6=-2\times3 - 6=-6 - 6=-12\), which is equal to the right - hand side.

• For \(x=-1\):

Substitute \(x=-1\) into the left - hand side of the original equation \(-2|8-(-1)|-6=-2\times|9|-6=-2\times9 - 6=-18 - 6=-24
eq - 12\)

Johan made a mistake when solving the second case of the absolute - value equation. When he solved \(8 - x=-3\), he should get \(x = 11\) instead of \(x=-1\) (a sign error occurred when moving terms around).

Answer:

He is not correct because he made a sign error.