We are tasked with solving the inequality:

\[
\frac{7(2x+1)}{5} > \frac{5(2x-1)}{3} + 4 - \frac{x}{2}
\]

Let's solve this step-by-step:

### Step 1: Eliminate the fractions
To eliminate the fractions, we will multiply through by the least common denominator (LCD) of 5, 3, and 2, which is 30.

Multiply every term by 30:

\[
30 \times \frac{7(2x+1)}{5} > 30 \times \frac{5(2x-1)}{3} + 30 \times 4 - 30 \times \frac{x}{2}
\]

Simplifying each term:

\[
6 \times 7(2x+1) > 10 \times 5(2x-1) + 120 - 15x
\]

\[
42(2x+1) > 50(2x-1) + 120 - 15x
\]

### Step 2: Expand the parentheses
Now, expand both sides:

\[
42(2x+1) = 42 \times 2x + 42 \times 1 = 84x + 42
\]

\[
50(2x-1) = 50 \times 2x - 50 \times 1 = 100x - 50
\]

Substituting back into the inequality:

\[
84x + 42 > 100x - 50 + 120 - 15x
\]

### Step 3: Simplify the inequality
Now, combine like terms on the right side:

\[
84x + 42 > (100x - 15x) + (-50 + 120)
\]

\[
84x + 42 > 85x + 70
\]

### Step 4: Collect like terms
Now, move all the terms involving \(x\) to one side and the constants to the other side:

\[
84x - 85x > 70 - 42
\]

\[
-x > 28
\]

### Step 5: Solve for \(x\)
To solve for \(x\), divide both sides by -1, remembering to reverse the inequality sign:

\[
x < -28
\]

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

\[
x < -28
\]

Would you like more details or further clarification?

Here are some related questions:
1. How do you solve linear inequalities involving fractions?
2. What are some common mistakes to avoid when solving inequalities?
3. How do you know when to reverse the inequality sign?
4. How can you verify the solution to an inequality?
5. Can inequalities have infinite solutions?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing by a negative number!

Learn how to solve the inequality 7(2x+1)/5 > 5(2x-1)/3 + 4 - x/2 with a detailed step-by-step solution. This problem covers algebraic concepts including solving linear inequalities involving fractions and properties of inequalities. Suitable for students in Grades 8-10.

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