The problem asks us to find the value of $x$ that satisfies the equation obtained by replacing the inequality sign in $\frac{-10+x}{4} + 5 \geq \frac{7x-5}{3}$ with an equal sign. In other words, we need to solve for $x$ in the equation $\frac{-10+x}{4} + 5 = \frac{7x-5}{3}$. Also, we need to explain how solving the equation $7x+5 = 2x+35$ helps solve the inequality $7x+5 > 2x+35$.

1. Problem Description

The problem asks us to find the value of $x$ that satisfies the equation obtained by replacing the inequality sign in $\frac{-10+x}{4} + 5 \geq \frac{7x-5}{3}$ with an equal sign. In other words, we need to solve for $x$ in the equation $\frac{-10+x}{4} + 5 = \frac{7x-5}{3}$. Also, we need to explain how solving the equation $7x+5 = 2x+35$ helps solve the inequality $7x+5 > 2x+35$.

2. Solution Steps

First, solve $\frac{-10+x}{4} + 5 = \frac{7x-5}{3}$.

Multiply both sides of the equation by 12 to eliminate fractions:

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

$3(-10+x) + 60 = 4(7x-5)$

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

$3x+30 = 28x-20$

Subtract $3x$ from both sides:

$30 = 25x-20$

Add 20 to both sides:

$50 = 25x$

Divide both sides by 25:

$x = \frac{50}{25} = 2$

Second, explain how solving the equation $7x+5=2x+35$ helps solve the inequality $7x+5>2x+35$. Solving $7x+5=2x+35$ yields $x=6$. This value, $x=6$, is the boundary between where $7x+5 > 2x+35$ and where $7x+5 < 2x+35$. To solve the inequality $7x+5 > 2x+35$, we solve the equation $7x+5=2x+35$ to find the boundary $x=6$. Then, we test a value of $x$ less than 6, such as $x=0$, and we have $7(0)+5 > 2(0)+35$, which simplifies to $5>35$, which is false. Then we test a value of $x$ greater than 6, such as $x=7$. We have $7(7)+5 > 2(7)+35$, which simplifies to $49+5 > 14+35$, or $54>49$, which is true. Therefore, the solution to the inequality $7x+5 > 2x+35$ is $x>6$.

3. Final Answer

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

Solving the equation $7x+5 = 2x+35$ finds the boundary point $x=6$ for the inequality $7x+5>2x+35$. We can then test values on either side of $x=6$ to determine the solution to the inequality.