Question

1. solve the radical equation: \\(\sqrt{5x - 11} = 7\\)

Explanation:

Step1: Square both sides to eliminate radical

To solve the radical equation \(\sqrt{5x - 11}=7\), we square both sides of the equation. Squaring the left side will eliminate the square root, and squaring the right side gives us a new equation.

$$ (\sqrt{5x - 11})^2 = 7^2 $$

Simplifying both sides, we get:

$$ 5x - 11 = 49 $$

Step2: Solve for x

Now we solve the linear equation \(5x - 11 = 49\) for \(x\). First, we add 11 to both sides of the equation to isolate the term with \(x\):

$$ 5x - 11 + 11 = 49 + 11 $$

Simplifying both sides, we have:

$$ 5x = 60 $$

Then, we divide both sides by 5 to solve for \(x\):

$$ x=\frac{60}{5} $$

Simplifying the right side, we get:

$$ x = 12 $$

Step3: Check the solution

We should check if \(x = 12\) is a valid solution by substituting it back into the original equation.
Left side: \(\sqrt{5(12)-11}=\sqrt{60 - 11}=\sqrt{49} = 7\)
Right side: \(7\)
Since both sides are equal, \(x = 12\) is a valid solution.

Answer:

\(x = 12\)