The problem states that Noah is trying to solve the inequality $7x + 5 > 2x + 35$. He first solves the equation $7x + 5 = 2x + 35$ and finds that $x = 6$. The question asks how this solution helps Noah solve the inequality $7x + 5 > 2x + 35$.

1. Problem Description

The problem states that Noah is trying to solve the inequality $7x + 5 > 2x + 35$. He first solves the equation $7x + 5 = 2x + 35$ and finds that $x = 6$. The question asks how this solution helps Noah solve the inequality $7x + 5 > 2x + 35$.

2. Solution Steps

The solution to the equation $7x + 5 = 2x + 35$, which is $x = 6$, represents the point where the expressions $7x + 5$ and $2x + 35$ are equal. This point divides the number line into two intervals.

To solve the inequality $7x + 5 > 2x + 35$, we need to determine for what values of $x$ the expression $7x + 5$ is greater than the expression $2x + 35$.

We know that at $x = 6$, the two expressions are equal. So we need to test values on either side of $x = 6$.

Let's test $x = 7$ (a value greater than 6):

$7(7) + 5 = 49 + 5 = 54$

$2(7) + 35 = 14 + 35 = 49$

Since $54 > 49$, the inequality holds for $x = 7$.

Let's test $x = 5$ (a value less than 6):

$7(5) + 5 = 35 + 5 = 40$

$2(5) + 35 = 10 + 35 = 45$

Since $40 < 45$, the inequality does not hold for $x = 5$.

Therefore, the solution to the inequality is $x > 6$. The value $x = 6$ is the boundary between the region where $7x+5 > 2x+35$ and the region where $7x+5 < 2x+35$. Solving the equation allows us to find this boundary point.

3. Final Answer

The solution to the equation $7x + 5 = 2x + 35$, which is $x=6$, helps Noah solve the inequality $7x + 5 > 2x + 35$ by giving him the boundary point. He knows that the solution to the inequality will either be $x > 6$ or $x < 6$. He can then test a value in each interval to determine which interval contains the solution to the inequality. In this case, the solution to the inequality is $x > 6$.