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Let's solve the inequality:

\[
2x + \frac{7}{2} \geq 29x - 10
\]

### Step 1: Eliminate the fraction
To eliminate the fraction, multiply the entire inequality by 2:

\[
2 \cdot \left(2x + \frac{7}{2}\right) \geq 2 \cdot (29x - 10)
\]

This simplifies to:

\[
4x + 7 \geq 58x - 20
\]

### Step 2: Move the variable terms to one side
Subtract \(4x\) from both sides to get the variables on one side:

\[
7 \geq 58x - 4x - 20
\]

This simplifies to:

\[
7 \geq 54x - 20
\]

### Step 3: Move the constant terms to the other side
Add 20 to both sides:

\[
7 + 20 \geq 54x
\]

This simplifies to:

\[
27 \geq 54x
\]

### Step 4: Solve for \(x\)
Divide both sides by 54:

\[
\frac{27}{54} \geq x
\]

This simplifies to:

\[
\frac{1}{2} \geq x
\]

or equivalently:

\[
x \leq \frac{1}{2}
\]

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

\[
x \leq \frac{1}{2}
\]

---

Would you like more details on any step? Here are some related questions you might want to explore:

1. What are the steps to solve inequalities involving fractions?
2. How does multiplying or dividing by a negative number affect inequalities?
3. Can the same process be applied to equations with two variables?
4. What is the difference between solving equations and inequalities?
5. How would this inequality change if the sign was reversed (e.g., \(2x + \frac{7}{2} \leq 29x - 10\))?

**Tip:** When solving inequalities, remember that multiplying or dividing both sides by a negative number reverses the inequality sign.

Learn how to solve the linear inequality 2x + 7/2 ≥ 29x - 10 with a detailed step-by-step solution. This problem covers key algebraic concepts, including manipulating equations with fractions and solving inequalities. Suitable for students in Grades 9-10.

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