https://file.aiquickdraw.com/mathbot/file/1726027424430jqjdyjk7.jpg

The equation in the image is:

\[
\left| \frac{x + 1}{5 - 2x} \right| = 3
\]

To solve this, we'll approach it by considering two cases for the absolute value expression.

### Case 1: 
When \(\frac{x + 1}{5 - 2x} = 3\)

\[
\frac{x + 1}{5 - 2x} = 3
\]

Multiply both sides by \(5 - 2x\) to eliminate the denominator (assuming \(5 - 2x \neq 0\)):

\[
x + 1 = 3(5 - 2x)
\]

Expand the right-hand side:

\[
x + 1 = 15 - 6x
\]

Now, bring all terms involving \(x\) to one side:

\[
x + 6x = 15 - 1
\]

\[
7x = 14
\]

Solve for \(x\):

\[
x = 2
\]

### Case 2: 
When \(\frac{x + 1}{5 - 2x} = -3\)

\[
\frac{x + 1}{5 - 2x} = -3
\]

Multiply both sides by \(5 - 2x\):

\[
x + 1 = -3(5 - 2x)
\]

Expand the right-hand side:

\[
x + 1 = -15 + 6x
\]

Now, bring all terms involving \(x\) to one side:

\[
x - 6x = -15 - 1
\]

\[
-5x = -16
\]

Solve for \(x\):

\[
x = \frac{16}{5}
\]

### Conclusion:
The solutions to the equation are:

\[
x = 2 \quad \text{or} \quad x = \frac{16}{5}
\]

Would you like more details or have any further questions?

Here are 5 related questions to deepen understanding:

1. How do you check whether each solution satisfies the original equation?
2. What is the significance of the absolute value in the equation?
3. How do we handle extraneous solutions when solving absolute value equations?
4. Could you explain how multiplying by the denominator works in this context?
5. What if the denominator in this problem became zero? How would the solution change?

**Tip:** When solving absolute value equations, always remember to consider both the positive and negative cases of the expression inside the absolute value.

Solve the equation: |(x + 1) / (5 - 2x)| = 3

Learn how to solve the absolute value equation |(x + 1) / (5 - 2x)| = 3 with a detailed step-by-step explanation. This problem explores absolute value equations and rational expressions, suitable for students in Grades 8-10.

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