Questions: Question 20 Solve the absolute value equation: -2-x+4-1=-19 -5 and 13 -5 and 5 -6 and 6 No Solution Submit and End

Solution Steps

To solve the absolute value equation \(-2|-x+4|-1=-19\), we first isolate the absolute value expression. Then, we solve the resulting equations by considering both the positive and negative scenarios of the expression inside the absolute value. Finally, we check if the solutions satisfy the original equation.

The given equation is:

\[ -2|-x+4|-1=-19 \]

First, we need to isolate the absolute value expression. Add 1 to both sides:

\[ -2|-x+4| = -18 \]

Next, divide both sides by -2:

\[ |-x+4| = 9 \]

The equation \(|-x+4| = 9\) implies two possible cases:

1. \(-x+4 = 9\)
2. \(-x+4 = -9\)

Subtract 4 from both sides:

\[ -x = 5 \]

Multiply both sides by -1:

\[ x = -5 \]

Subtract 4 from both sides:

\[ -x = -13 \]

Multiply both sides by -1:

\[ x = 13 \]

We have two potential solutions: \(x = -5\) and \(x = 13\). Let's verify them in the original equation:

Substitute \(x = -5\) into the original equation:

\[ -2|-(-5)+4|-1 = -19 \]

Simplify:

\[ -2|5+4|-1 = -19 \]

\[ -2|9|-1 = -19 \]

\[ -18-1 = -19 \]

This is true, so \(x = -5\) is a valid solution.

Substitute \(x = 13\) into the original equation:

\[ -2|-(13)+4|-1 = -19 \]

Simplify:

\[ -2|-13+4|-1 = -19 \]

\[ -2|-9|-1 = -19 \]

\[ -18-1 = -19 \]

This is true, so \(x = 13\) is also a valid solution.

Final Answer