`(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: Find a common denominator for the right-hand side.

The denominators are 3 and 5. The least common multiple (LCM) of 3 and 5 is 15. We will rewrite the right-hand side with a 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 inequality:

\[ \frac{x}{2} \geq \frac{25x - 10 - (21x - 9)}{15} \]

### Step 2: Simplify the right-hand side.

Combine the terms in the numerator:

\[ 25x - 10 - 21x + 9 = (25x - 21x) + (-10 + 9) = 4x - 1 \]

So the inequality now looks like this:

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

### Step 3: Eliminate the fractions by multiplying through by the least common multiple (LCM).

The LCM of 2 and 15 is 30. Multiply both sides of the inequality by 30:

\[ 30 \cdot \frac{x}{2} \geq 30 \cdot \frac{4x - 1}{15} \]

This simplifies to:

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

### Step 4: Distribute on the right-hand side.

Distributing the 2 gives:

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

### Step 5: Move all terms involving \(x\) to one side.

Subtract \(8x\) from both sides:

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

This simplifies to:

\[ 7x \geq -2 \]

### Step 6: Solve for \(x\).

Divide both sides by 7:

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

### Step 7: Write the solution in interval notation.

The solution can be expressed in interval notation as:

\[ x \in \left[-\frac{2}{7}, \infty\right) \]

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