https://file.aiquickdraw.com/mathbot/file/1726649295132tm0g06mg.jpg

The inequality in the image is:

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

Let's solve this step by step.

### Step 1: Eliminate the fractions by multiplying both sides by \((x + 1)(3x - 1)\), the common denominator.
We must be cautious about the restrictions on \(x\), as the denominators cannot be zero:
- \(x + 1 \neq 0 \Rightarrow x \neq -1\)
- \(3x - 1 \neq 0 \Rightarrow x \neq \frac{1}{3}\)

Multiplying both sides by \((x + 1)(3x - 1)\):

\[
1(3x - 1) < 2(x + 1)
\]

### Step 2: Expand both sides.

On the left side:
\[
3x - 1
\]

On the right side:
\[
2(x + 1) = 2x + 2
\]

So, the inequality becomes:
\[
3x - 1 < 2x + 2
\]

### Step 3: Solve for \(x\).

Subtract \(2x\) from both sides:
\[
x - 1 < 2
\]

Add 1 to both sides:
\[
x < 3
\]

### Step 4: Combine with domain restrictions.

From earlier, we have:
- \(x \neq -1\)
- \(x \neq \frac{1}{3}\)

Thus, the solution is:

\[
x < 3, \quad x \neq -1, \quad x \neq \frac{1}{3}
\]

### Final Answer:
The solution to the inequality is:
\[
x \in (-\infty, -1) \cup \left(-1, \frac{1}{3}\right) \cup \left(\frac{1}{3}, 3\right)
\]

Do you have any questions, or would you like more details?

Here are 5 related questions:
1. How do you solve inequalities with fractions?
2. What are the general rules for solving linear inequalities?
3. How do you handle cases where the variable appears in the denominator?
4. What are the steps to graph a solution set on a number line?
5. How do restrictions on the domain affect the solution?

**Tip:** When solving inequalities, always remember to check for restrictions due to denominators or square roots.

Solve the inequality: 1/(x + 1) < 2/(3x - 1)

Learn how to solve the rational inequality 1/(x + 1) < 2/(3x - 1) with step-by-step instructions. This problem involves algebraic manipulation, cross multiplication, and handling domain restrictions. Suitable for students in Grades 9-12.

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