To solve the equation [tex]\(\sqrt{6x + 46} - 18 = -16\)[/tex] for [tex]\(x\)[/tex], follow these steps:
1. Isolate the square root term:
[tex]\[
\sqrt{6x + 46} - 18 = -16
\][/tex]
Add 18 to both sides of the equation:
[tex]\[
\sqrt{6x + 46} = -16 + 18
\][/tex]
Simplify the right side:
[tex]\[
\sqrt{6x + 46} = 2
\][/tex]
2. Eliminate the square root by squaring both sides:
[tex]\[
(\sqrt{6x + 46})^2 = 2^2
\][/tex]
This simplifies to:
[tex]\[
6x + 46 = 4
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Subtract 46 from both sides:
[tex]\[
6x = 4 - 46
\][/tex]
Simplify the right side:
[tex]\[
6x = -42
\][/tex]
Divide both sides by 6:
[tex]\[
x = \frac{-42}{6}
\][/tex]
Simplify the fraction:
[tex]\[
x = -7
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
\boxed{-7}
\][/tex]
