To solve the equation [tex]\( 3 + \sqrt{2x + 4} = 7 \)[/tex], let's walk through the steps to isolate and solve for [tex]\( x \)[/tex]:
1. Isolate the square root term:
[tex]\[
3 + \sqrt{2x + 4} = 7
\][/tex]
Subtract 3 from both sides to isolate the square root:
[tex]\[
\sqrt{2x + 4} = 7 - 3
\][/tex]
Simplify the right side:
[tex]\[
\sqrt{2x + 4} = 4
\][/tex]
2. Eliminate the square root by squaring both sides:
[tex]\[
(\sqrt{2x + 4})^2 = 4^2
\][/tex]
This leads to:
[tex]\[
2x + 4 = 16
\][/tex]
3. Solve the resulting linear equation:
Subtract 4 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
2x = 16 - 4
\][/tex]
Simplify the right side:
[tex]\[
2x = 12
\][/tex]
Finally, divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{12}{2} = 6
\][/tex]
Thus, the solution to the equation [tex]\( 3 + \sqrt{2x + 4} = 7 \)[/tex] is [tex]\(\boxed{6}\)[/tex]. The correct answer is option C: [tex]\( x = 6 \)[/tex].
