Certainly! Let's solve the given equation step-by-step:
We start with the equation:
[tex]\[
\frac{1}{3 \sqrt{2x - 3}} = 2
\][/tex]
Step 1: Isolate the square root term
To isolate the square root term, we first want to eliminate the fraction. Multiply both sides of the equation by [tex]\(3 \sqrt{2x - 3}\)[/tex]:
[tex]\[
1 = 6 \sqrt{2x - 3}
\][/tex]
Step 2: Solve for the square root term
Next, we’ll divide both sides by 6 to solve for the square root term:
[tex]\[
\frac{1}{6} = \sqrt{2x - 3}
\][/tex]
Step 3: Eliminate the square root
Now we need to eliminate the square root by squaring both sides of the equation:
[tex]\[
\left( \frac{1}{6} \right)^2 = ( \sqrt{2x - 3} )^2
\][/tex]
This simplifies to:
[tex]\[
\frac{1}{36} = 2x - 3
\][/tex]
Step 4: Solve the linear equation for [tex]\(x\)[/tex]
We now have a simple linear equation. Begin by adding 3 to both sides to isolate the [tex]\(2x\)[/tex] term:
[tex]\[
\frac{1}{36} + 3 = 2x
\][/tex]
Convert [tex]\(3\)[/tex] to a fraction with a denominator of 36:
[tex]\[
\frac{1}{36} + \frac{108}{36} = \frac{109}{36} = 2x
\][/tex]
Now divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{109}{72}
\][/tex]
Thus, the solution to the equation [tex]\(\frac{1}{3 \sqrt{2x-3}}=2\)[/tex] is:
[tex]\[
x = \frac{109}{72}
\][/tex]
