The problem asks to find the value of $x$ that makes the two sides of the inequality $\frac{-10+x}{4}+5 \ge \frac{7x-5}{3}$ equal. This means we need to solve the equation $\frac{-10+x}{4}+5 = \frac{7x-5}{3}$.

1. Problem Description

The problem asks to find the value of $x$ that makes the two sides of the inequality $\frac{-10+x}{4}+5 \ge \frac{7x-5}{3}$ equal. This means we need to solve the equation $\frac{-10+x}{4}+5 = \frac{7x-5}{3}$.

2. Solution Steps

To solve the equation, we first multiply both sides by 12 (the least common multiple of 4 and 3) to eliminate the fractions:

$12 \cdot (\frac{-10+x}{4}+5) = 12 \cdot \frac{7x-5}{3}$

$3(-10+x) + 12(5) = 4(7x-5)$

$-30 + 3x + 60 = 28x - 20$

$3x + 30 = 28x - 20$

Now, we want to isolate $x$. Subtract $3x$ from both sides:

$30 = 25x - 20$

Add 20 to both sides:

$50 = 25x$

Finally, divide both sides by 25:

$x = \frac{50}{25}$

$x = 2$

3. Final Answer

The value of $x$ that produces equality is $x = 2$.