Question

4.) \\(\sqrt{5x + 3} = \sqrt{4x + 5}\\)

Explanation:

Step1: Square both sides

To eliminate the square roots, we square both sides of the equation \(\sqrt{5x + 3} = \sqrt{4x + 5}\). Squaring a square root cancels out the square root, so we get:
\((\sqrt{5x + 3})^2 = (\sqrt{4x + 5})^2\)
Which simplifies to:
\(5x + 3 = 4x + 5\)

Step2: Solve for x

Subtract \(4x\) from both sides:
\(5x - 4x + 3 = 4x - 4x + 5\)
\(x + 3 = 5\)
Then subtract 3 from both sides:
\(x + 3 - 3 = 5 - 3\)
\(x = 2\)

Step3: Check the solution

We need to check if \(x = 2\) makes the original equation true. Substitute \(x = 2\) into the original equation:
Left side: \(\sqrt{5(2) + 3} = \sqrt{10 + 3} = \sqrt{13}\)
Right side: \(\sqrt{4(2) + 5} = \sqrt{8 + 5} = \sqrt{13}\)
Since both sides are equal, \(x = 2\) is a valid solution.

Answer:

\(x = 2\)