a) Solve the inequality:
𝑥
3
>
𝑥
2
+ 1.

To solve the inequality \(\frac{x}{3} > \frac{x}{2} + 1\), follow these steps:

### Step 1: Eliminate the fractions
Multiply both sides of the inequality by the least common denominator (LCD) of the fractions, which is 6 in this case:

\[
6 \cdot \left(\frac{x}{3}\right) > 6 \cdot \left(\frac{x}{2} + 1\right)
\]

This simplifies to:

\[
2x > 3x + 6
\]

### Step 2: Simplify the inequality
Next, subtract \(3x\) from both sides to isolate the variable \(x\) on one side:

\[
2x - 3x > 6
\]

\[
-x > 6
\]

### Step 3: Solve for \(x\)
To solve for \(x\), multiply both sides by \(-1\). Remember, when you multiply or divide an inequality by a negative number, you must reverse the inequality sign:

\[
x < -6
\]

### Final Answer:
The solution to the inequality is:

\[
x < -6
\]

This means \(x\) must be less than \(-6\) to satisfy the inequality.

---

Would you like more details or have any questions about this solution? Here are some related questions for further practice:

1. Solve the inequality \(\frac{2x}{5} > \frac{x}{3} + 4\).
2. Solve the inequality \(4x - 3 > 2x + 7\).
3. Solve the inequality \(3 - 2x \leq 5 + x\).
4. Solve the inequality \(\frac{2x + 1}{3} < \frac{4x - 2}{5}\).
5. Solve the inequality \(x^2 - 4x + 3 \geq 0\).

**Tip:** Always remember to reverse the inequality sign when you multiply or divide by a negative number!

Learn how to solve the inequality x/3 > x/2 + 1 with detailed steps and explanations. This problem covers inequalities and fraction operations, suitable for students in Grades 9-12.

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