Question

1. solve for x: (sqrt{3x + 45} + 1 = 7)

Explanation:

Step 1: Isolate the square root term

Subtract 1 from both sides of the equation \(\sqrt{3x + 45}+1 = 7\) to get \(\sqrt{3x + 45}=7 - 1\).
Simplifying the right side, we have \(\sqrt{3x + 45}=6\).

Step 2: Eliminate the square root

Square both sides of the equation \(\sqrt{3x + 45}=6\) to obtain \((\sqrt{3x + 45})^2=6^2\).
This simplifies to \(3x + 45 = 36\).

Step 3: Solve for x

Subtract 45 from both sides: \(3x=36 - 45\).
Simplifying the right side gives \(3x=-9\).
Divide both sides by 3: \(x=\frac{-9}{3}=- 3\).

Step 4: Check the solution

Substitute \(x = - 3\) back into the original equation:
Left side: \(\sqrt{3(-3)+45}+1=\sqrt{-9 + 45}+1=\sqrt{36}+1 = 6 + 1=7\), which equals the right side. So the solution is valid.

Answer:

\(x=-3\)