To solve the equation

$-|7 - 2x| + 11 = -8,$

we need to isolate the absolute value expression and then handle the two possible cases (positive and negative scenarios inside the absolute value). Here's a step-by-step breakdown:

Step 1: Isolate the Absolute Value

Start by moving 11 to the other side:

$-|7 - 2x| = -8 - 11,$

$-|7 - 2x| = -19.$

Step 2: Remove the Negative Sign

Multiply both sides by -1 to get rid of the negative sign in front of the absolute value:

$|7 - 2x| = 19.$

Step 3: Split into Two Cases

The absolute value equation $|7 - 2x| = 19$ implies two cases:

Case 1: $7 - 2x = 19$

Solve for $x$:

$7 - 2x = 19,$

$-2x = 19 - 7,$

$-2x = 12,$

$x = -6.$

Case 2: $7 - 2x = -19$

Solve for $x$:

$7 - 2x = -19,$

$-2x = -19 - 7,$

$-2x = -26,$

$x = 13.$

Final Solution

The solutions to the equation are:

$x = -6 \quad \text{and} \quad x = 13.$

Would you like a more detailed explanation or have any questions?

Related Questions

1. How do you solve equations involving absolute values with more complex expressions?
2. What happens if the absolute value is equal to a negative number?
3. Can you solve the equation if there are variables inside the absolute value?
4. How do you graph equations involving absolute values?
5. How can you check your solutions for absolute value equations?

Tip

Always check the solutions in the original equation to ensure that they satisfy all conditions, especially when dealing with absolute values and potential extraneous solutions.