`(x)/(2) ge ((5x - 2))/(3) - ((7x - 3))/(5) `

Explanation

To solve the inequality \(\frac{x}{2} \geq \frac{5x - 2}{3} - \frac{7x - 3}{5}\), we will follow these steps:

### Step 1: Simplify the Right-Hand Side
First, we need to simplify the right-hand side of the inequality. The right-hand side consists of two fractions, so we will find a common denominator.

The denominators are 3 and 5, and the least common multiple (LCM) of 3 and 5 is 15. We will rewrite each term with the common denominator of 15:

\[ \frac{5x - 2}{3} = \frac{5(5x - 2)}{15} = \frac{25x - 10}{15} \]
\[ \frac{7x - 3}{5} = \frac{3(7x - 3)}{15} = \frac{21x - 9}{15} \]

Now, we can rewrite the right-hand side:

\[ \frac{5x - 2}{3} - \frac{7x - 3}{5} = \frac{25x - 10 - (21x - 9)}{15} = \frac{25x - 10 - 21x + 9}{15} = \frac{4x - 1}{15} \]

### Step 2: Rewrite the Inequality
Now we can rewrite the original inequality:

\[ \frac{x}{2} \geq \frac{4x - 1}{15} \]

### Step 3: Cross-Multiply
To eliminate the fractions, we will cross-multiply. Note that since both sides are positive, we can do this without reversing the inequality:

\[ 15x \geq 2(4x - 1) \]

### Step 4: Distribute and Simplify
Now, we will distribute on the right-hand side:

\[ 15x \geq 8x - 2 \]

Next, we will move all terms involving \(x\) to one side of the inequality:

\[ 15x - 8x \geq -2 \]

This simplifies to:

\[ 7x \geq -2 \]

### Step 5: Solve for \(x\)
Now, divide both sides by 7:

\[ x \geq -\frac{2}{7} \]

### Step 6: Conclusion
The solution to the inequality is:

\[ x \geq -\frac{2}{7} \]

### Step 7: Graph the Solution
To graph this solution on a number line, we will mark the point \(-\frac{2}{7}\) with a closed dot (since the inequality includes equality) and shade to the right to indicate all values greater than or equal to \(-\frac{2}{7}\).

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