To solve the equation [tex]\(\sqrt{2x - 3} + 4 = 7\)[/tex], let's go through the process step by step:
1. Isolate the square root term:
[tex]\[\sqrt{2x - 3} + 4 = 7\][/tex]
Subtract 4 from both sides to isolate the square root term:
[tex]\[\sqrt{2x - 3} = 7 - 4\][/tex]
[tex]\[\sqrt{2x - 3} = 3\][/tex]
2. Eliminate the square root:
To eliminate the square root, square both sides of the equation:
[tex]\[(\sqrt{2x - 3})^2 = 3^2\][/tex]
[tex]\(2x - 3 = 9\)[/tex]
3. Solve for [tex]\(x\)[/tex]:
Add 3 to both sides to solve for [tex]\(2x\)[/tex]:
[tex]\(2x - 3 + 3 = 9 + 3\)[/tex]
[tex]\(2x = 12\)[/tex]
Now, divide both sides by 2 to find [tex]\(x\)[/tex]:
[tex]\(x = \frac{12}{2}\)[/tex]
[tex]\(x = 6\)[/tex]
4. Verify the solution:
Substitute [tex]\(x = 6\)[/tex] back into the original equation to check if it satisfies the equation:
[tex]\[\sqrt{2(6) - 3} + 4 = 7\][/tex]
[tex]\[\sqrt{12 - 3} + 4 = 7\][/tex]
[tex]\[\sqrt{9} + 4 = 7\][/tex]
[tex]\[3 + 4 = 7\][/tex]
[tex]\[7 = 7\][/tex]
Since the equation is satisfied, the solution [tex]\(x = 6\)[/tex] is correct.
Therefore, the answer is:
[tex]\[ \boxed{x = 6} \][/tex]
