To solve the equation [tex]\(\sqrt{11x + 5} = 7\)[/tex], let's follow a systematic approach:
### Step 1: Isolate the square root
The equation we start with is:
[tex]\[
\sqrt{11x + 5} = 7
\][/tex]
### Step 2: Eliminate the square root
To eliminate the square root, we square both sides of the equation:
[tex]\[
(\sqrt{11x + 5})^2 = 7^2
\][/tex]
This simplifies to:
[tex]\[
11x + 5 = 49
\][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
Next, isolate [tex]\(x\)[/tex] by performing algebraic operations:
[tex]\[
11x + 5 = 49
\][/tex]
Subtract 5 from both sides of the equation:
[tex]\[
11x = 44
\][/tex]
Divide both sides by 11:
[tex]\[
x = 4
\][/tex]
### Step 4: Verify the solution
To ensure the solution [tex]\( x = 4 \)[/tex] is not extraneous, we substitute [tex]\( x = 4 \)[/tex] back into the original equation and check if it holds true:
[tex]\[
\sqrt{11(4) + 5} = 7
\][/tex]
Simplify inside the square root:
[tex]\[
\sqrt{44 + 5} = 7
\][/tex]
[tex]\[
\sqrt{49} = 7
\][/tex]
Since [tex]\(\sqrt{49} = 7\)[/tex], the original equation is satisfied, confirming that our solution is correct.
### Conclusion
The solution to the equation [tex]\(\sqrt{11x + 5} = 7\)[/tex] is:
[tex]\[
x = 4
\][/tex]
This solution is not extraneous, as substituting it back into the original equation verifies that it satisfies the equation.
