Solve for all values of xx in simplest form.
minus, vertical bar, 1, minus, 5, x, vertical bar, plus, 3, equals, minus, 6
−∣1−5x∣+3=
−6

We are given the equation:

\[
- |1 - 5x| + 3 = -6
\]

Let's solve for \( x \) step by step.

### Step 1: Isolate the absolute value term
Subtract 3 from both sides:

\[
- |1 - 5x| = -6 - 3
\]

Simplify the right-hand side:

\[
- |1 - 5x| = -9
\]

### Step 2: Remove the negative sign
Multiply both sides of the equation by \( -1 \) to eliminate the negative sign:

\[
|1 - 5x| = 9
\]

### Step 3: Solve the absolute value equation
The equation \( |1 - 5x| = 9 \) means that \( 1 - 5x \) can either be 9 or \( -9 \). So, we have two cases to consider:

#### Case 1: \( 1 - 5x = 9 \)
Solve for \( x \):

\[
1 - 5x = 9
\]

Subtract 1 from both sides:

\[
-5x = 8
\]

Now, divide by \( -5 \):

\[
x = -\frac{8}{5}
\]

#### Case 2: \( 1 - 5x = -9 \)
Solve for \( x \):

\[
1 - 5x = -9
\]

Subtract 1 from both sides:

\[
-5x = -10
\]

Now, divide by \( -5 \):

\[
x = 2
\]

### Final Answer:
The solutions are:

\[
x = -\frac{8}{5} \quad \text{or} \quad x = 2
\]

Do you have any questions or would you like further details?

Here are 5 related questions to expand your understanding:
1. How do you solve absolute value inequalities?
2. What happens if the absolute value term is equal to zero?
3. How would you graph the solutions of this equation?
4. Can absolute value equations have no solution?
5. How do you solve absolute value equations with variables on both sides?

**Tip:** Always remember that an absolute value equation can have two possible cases—positive and negative—when isolating the absolute value term.

Solve for all values of x in simplest form. −|1 − 5x| + 3 = −6

Discover how to solve the absolute value equation −|1 − 5x| + 3 = −6 with a clear, step-by-step breakdown. Learn about key mathematical concepts like absolute value equations and linear equations, using the absolute value theorem. Suitable for students in Grades 8-10.

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