Noah is solving the inequality $7x + 5 > 2x + 35$. He first solves the equation $7x + 5 = 2x + 35$ and finds $x = 6$. The question is how this helps him solve the original inequality.

1. Problem Description

Noah is solving the inequality $7x + 5 > 2x + 35$. He first solves the equation $7x + 5 = 2x + 35$ and finds $x = 6$. The question is how this helps him solve the original inequality.

2. Solution Steps

Solving the equation $7x + 5 = 2x + 35$ gives the point where the expressions $7x + 5$ and $2x + 35$ are equal. This is the boundary point.

To solve the inequality $7x + 5 > 2x + 35$, we know that $x = 6$ is the boundary point.

We need to test values on either side of $x = 6$ to determine the solution to the inequality.

Test $x = 5$:

$7(5) + 5 > 2(5) + 35$

$35 + 5 > 10 + 35$

$40 > 45$ (False)

Test $x = 7$:

$7(7) + 5 > 2(7) + 35$

$49 + 5 > 14 + 35$

$54 > 49$ (True)

Therefore, the solution to the inequality is $x > 6$.

3. Final Answer

The solution to the equation $7x + 5 = 2x + 35$ gives the boundary point $x=6$. We test values on either side of $x=6$ to find the solution to the inequality $7x + 5 > 2x + 35$. The solution is $x > 6$.