To solve the equation [tex]\(\sqrt{2x - 3} + 4 = 7\)[/tex], follow these steps:
1. Isolate the square root term:
We start by isolating [tex]\(\sqrt{2x - 3}\)[/tex] on one side of the equation.
[tex]\[
\sqrt{2x - 3} + 4 = 7
\][/tex]
Subtract 4 from both sides:
[tex]\[
\sqrt{2x - 3} = 3
\][/tex]
2. Eliminate the square root:
Square both sides of the equation to remove the square root.
[tex]\[
(\sqrt{2x - 3})^2 = 3^2
\][/tex]
This simplifies to:
[tex]\[
2x - 3 = 9
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Rearrange the equation to solve for [tex]\(x\)[/tex].
[tex]\[
2x - 3 = 9
\][/tex]
Add 3 to both sides:
[tex]\[
2x = 12
\][/tex]
Divide by 2:
[tex]\[
x = 6
\][/tex]
4. Verify the solution:
Substitute [tex]\(x = 6\)[/tex] back into the original equation to ensure it is correct.
[tex]\[
\sqrt{2 \cdot 6 - 3} + 4 = 7
\][/tex]
Simplify inside the square root:
[tex]\[
\sqrt{12 - 3} + 4 = 7
\][/tex]
[tex]\[
\sqrt{9} + 4 = 7
\][/tex]
[tex]\[
3 + 4 = 7
\][/tex]
[tex]\[
7 = 7
\][/tex]
The left-hand side equals the right-hand side, confirming that [tex]\(x = 6\)[/tex] is indeed a solution.
So, the solution to the equation [tex]\(\sqrt{2x - 3} + 4 = 7\)[/tex] is [tex]\(x = 6\)[/tex]. Therefore, the correct answer is:
[tex]\[
x = 6
\][/tex]
