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: Rewrite the inequality
Start with the original inequality:
\[ \frac{x}{2} \geq \frac{5x - 8}{3} - \frac{7x - 13}{5} \]

### Step 2: Find a common denominator
The common denominator for the fractions on the right side is 15. We will rewrite each term with this denominator:
\[ \frac{x}{2} \geq \frac{5(5x - 8)}{15} - \frac{3(7x - 13)}{15} \]

### Step 3: Simplify the right side
Now, simplify the right side:
\[ \frac{x}{2} \geq \frac{25x - 40 - 21x + 39}{15} \]
Combine like terms:
\[ \frac{x}{2} \geq \frac{4x - 1}{15} \]

### Step 4: Eliminate the fractions
To eliminate the fractions, multiply both sides by 30 (the least common multiple of 2 and 15):
\[ 30 \cdot \frac{x}{2} \geq 30 \cdot \frac{4x - 1}{15} \]
This simplifies to:
\[ 15x \geq 8(4x - 1) \]

### Step 5: Distribute on the right side
Distributing the 8 on the right side gives:
\[ 15x \geq 32x - 8 \]

### Step 6: Rearrange the inequality
Now, move all terms involving \( x \) to one side:
\[ 15x - 32x \geq -8 \]
This simplifies to:
\[ -17x \geq -8 \]

### Step 7: Divide by -17
When dividing by a negative number, remember to reverse the inequality:
\[ x \leq \frac{-8}{-17} \]
This simplifies to:
\[ x \leq \frac{8}{17} \]

### Step 8: Conclusion
The solution to the inequality is:
\[ x \leq \frac{8}{17} \]
In interval notation, this is:
\[ (-\infty, \frac{8}{17}] \]

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