Solve : ` (x)/(2) ge (5x -8)/(3)-(7x-13)/(5)`

Explanation

To solve the inequality \( \frac{x}{2} \geq \frac{5x - 8}{3} - \frac{7x - 13}{5} \), we will follow these steps:

### Step 1: Find a common denominator
The denominators in the inequality are 2, 3, and 5. The least common multiple (LCM) of these numbers is 30. We will multiply every term by 30 to eliminate the fractions.

\[ 30 \cdot \frac{x}{2} \geq 30 \cdot \left( \frac{5x - 8}{3} - \frac{7x - 13}{5} \right) \]

### Step 2: Simplify each term
Now we simplify each term:

\[ 15x \geq 10(5x - 8) - 6(7x - 13) \]

### Step 3: Distribute and simplify
Distributing the constants inside the parentheses:

\[ 15x \geq 50x - 80 - 42x + 78 \]

### Step 4: Combine like terms
Now combine the like terms on the right side:

\[ 15x \geq (50x - 42x) + (-80 + 78) \]
\[ 15x \geq 8x - 2 \]

### Step 5: Isolate x
Now, we want to isolate \( x \). We can do this by subtracting \( 8x \) from both sides:

\[ 15x - 8x \geq -2 \]
\[ 7x \geq -2 \]

### Step 6: Divide by 7
Next, we divide both sides by 7:

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

### Step 7: Write the final solution
The solution to the inequality is:

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