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The inequality provided is:

\[
\frac{4x - 1}{6} + \frac{2x + 3}{5} \geq \frac{7x - 2}{3}
\]

To solve for \(x\), follow these steps:

### Step 1: Eliminate the fractions
To eliminate the denominators, find the least common multiple (LCM) of 6, 5, and 3, which is 30. Multiply each term by 30:

\[
30 \cdot \frac{4x - 1}{6} + 30 \cdot \frac{2x + 3}{5} \geq 30 \cdot \frac{7x - 2}{3}
\]

This simplifies to:

\[
5(4x - 1) + 6(2x + 3) \geq 10(7x - 2)
\]

### Step 2: Distribute and simplify
Now, distribute the numbers inside each term:

\[
20x - 5 + 12x + 18 \geq 70x - 20
\]

Combine like terms on the left side:

\[
32x + 13 \geq 70x - 20
\]

### Step 3: Isolate the variable \(x\)
Subtract \(32x\) from both sides:

\[
13 \geq 38x - 20
\]

Add 20 to both sides:

\[
33 \geq 38x
\]

### Step 4: Solve for \(x\)
Divide both sides by 38:

\[
\frac{33}{38} \geq x
\]

Thus, the solution is:

\[
x \leq \frac{33}{38}
\]

Or in decimal form:

\[
x \leq 0.8684
\]

Would you like any further details or have any questions about this solution?

Here are 5 related questions to explore:

1. How do you solve an inequality with variables on both sides?
2. What is the process for finding the least common multiple (LCM) of several numbers?
3. How does multiplying through by the LCM help in solving inequalities?
4. What should you be careful of when solving inequalities involving fractions?
5. How can the solution to an inequality be represented on a number line?

**Tip:** When multiplying or dividing both sides of an inequality by a negative number, always remember to reverse the inequality sign!

Learn how to solve the inequality involving fractions step-by-step. This problem covers inequalities, fractions, and equations, focusing on finding the least common multiple (LCM). Suitable for high school students.

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