To solve the equation

$|-9x - 5| - 6 = 2,$

follow these steps:

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Step 1: Isolate the absolute value term

Add 6 to both sides to isolate the absolute value:

$|-9x - 5| = 8.$

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Step 2: Solve for the two cases of the absolute value

The absolute value equation $|-9x - 5| = 8$ splits into two cases:

Case 1:

$-9x - 5 = 8$

Solve for $x$: $-9x = 8 + 5 \quad \Rightarrow \quad -9x = 13 \quad \Rightarrow \quad x = -\frac{13}{9}.$

Case 2:

$-9x - 5 = -8$

Solve for $x$: $-9x = -8 + 5 \quad \Rightarrow \quad -9x = -3 \quad \Rightarrow \quad x = \frac{1}{3}.$

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Step 3: Verify the solutions

Plug $x = -\frac{13}{9}$ and $x = \frac{1}{3}$ back into the original equation to confirm they satisfy it.

1. For $x = -\frac{13}{9}$: $|-9(-\frac{13}{9}) - 5| - 6 = |13 - 5| - 6 = |8| - 6 = 2 \quad \text{(Valid)}.$

2. For $x = \frac{1}{3}$: $|-9(\frac{1}{3}) - 5| - 6 = |-3 - 5| - 6 = |-8| - 6 = 2 \quad \text{(Valid)}.$

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Final Answer:

The solutions are: $x = -\frac{13}{9} \quad \text{and} \quad x = \frac{1}{3}.$

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Would you like further explanation or a graph of this equation?

Related Questions:

1. How do you graph an absolute value equation like this one?
2. What is the significance of breaking absolute value equations into two cases?
3. How do absolute value inequalities differ from equations in solving?
4. Can this equation have no solutions under different conditions?
5. What are some practical applications of absolute value equations?

Tip:

Always verify potential solutions when working with absolute value equations by substituting them into the original equation.