Let's solve the equation [tex]\(\sqrt{18x + 5} = 7\)[/tex] step-by-step.
### Step 1: Isolate the Square Root
The given equation is:
[tex]\[
\sqrt{18x + 5} = 7
\][/tex]
### Step 2: Square Both Sides to Eliminate the Square Root
To remove the square root, square both sides of the equation:
[tex]\[
(\sqrt{18x + 5})^2 = 7^2
\][/tex]
This simplifies to:
[tex]\[
18x + 5 = 49
\][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
Now, solve the resulting linear equation for [tex]\(x\)[/tex]:
1. First, isolate the term containing [tex]\(x\)[/tex] by subtracting 5 from both sides:
[tex]\[
18x + 5 - 5 = 49 - 5
\][/tex]
Simplifying this, we get:
[tex]\[
18x = 44
\][/tex]
2. Next, solve for [tex]\(x\)[/tex] by dividing both sides by 18:
[tex]\[
x = \frac{44}{18}
\][/tex]
3. Simplify the fraction:
[tex]\[
x = \frac{22}{9}
\][/tex]
Thus, the candidate solution is:
[tex]\[
x = \frac{22}{9}
\][/tex]
### Step 4: Check for Extraneous Solutions
To determine if this solution is extraneous, substitute [tex]\(x = \frac{22}{9}\)[/tex] back into the original equation [tex]\(\sqrt{18x + 5} = 7\)[/tex] and verify if both sides are equal.
Substitute [tex]\(x = \frac{22}{9}\)[/tex] into the left-hand side:
[tex]\[
\sqrt{18 \left(\frac{22}{9}\right) + 5}
\][/tex]
Calculate inside the square root:
[tex]\[
18 \left(\frac{22}{9}\right) = 2 \times 22 = 44
\][/tex]
So, the left-hand side becomes:
[tex]\[
\sqrt{44 + 5} = \sqrt{49} = 7
\][/tex]
Since both sides of the original equation are equal, [tex]\(x = \frac{22}{9}\)[/tex] is not an extraneous solution.
### Conclusion
The solution to the equation [tex]\(\sqrt{18x + 5} = 7\)[/tex] is:
[tex]\[
x = \frac{22}{9}
\][/tex]
This solution is valid and not extraneous, as verified by substitution back into the original equation.
