Question

find the real solutions of the equation.
\\(\sqrt{3x - 2} = 4\\)

what is the solution set? select the correct choice below and fill in any answer boxes in your choice.
\\(\bigcirc\\) a. \\(\\{\\ \\}\\)
(simplify your answer. use a comma to separate answers as needed.)
\\(\bigcirc\\) b. there are no real solutions.

Explanation:

Step1: Square both sides to eliminate the square root

To solve the equation \(\sqrt{3x - 2}=4\), we square both sides of the equation. Squaring the left side \((\sqrt{3x - 2})^2\) gives \(3x - 2\), and squaring the right side \(4^2\) gives \(16\). So we get the equation \(3x - 2 = 16\).

Step2: Solve for x

Add 2 to both sides of the equation \(3x - 2 = 16\). Adding 2 to the left side cancels out the -2, and adding 2 to the right side gives \(16 + 2 = 18\). So we have \(3x=18\). Then divide both sides by 3. Dividing the left side by 3 gives \(x\), and dividing the right side \(18\) by 3 gives \(6\). So \(x = 6\).

Step3: Check the solution

We need to check if \(x = 6\) satisfies the original equation. Substitute \(x = 6\) into the left side of the original equation \(\sqrt{3x - 2}\). We get \(\sqrt{3\times6 - 2}=\sqrt{18 - 2}=\sqrt{16}=4\), which is equal to the right side of the original equation. So \(x = 6\) is a valid solution.

Answer:

\(6\)