To solve this equation, we need to isolate the square root terms and then square both sides of the equation to eliminate the square roots.

Firstly, let's move the square root term to one side:

[tex]\sqrt{x+5} - \sqrt{x} = 1[/tex]
[tex]\sqrt{x+5} = 1 + \sqrt{x}[/tex]

Next, square both sides:

tex^2 = (1 + \sqrt{x})^2[/tex]
[tex]x + 5 = 1 + 2\sqrt{x} + x[/tex]

Now, simplify the equation:

[tex]4 = 2\sqrt{x}[/tex]
[tex]2 = \sqrt{x}[/tex]

Finally, square both sides again to get the value of x:

[tex]4 = x[/tex]

Therefore, the solution to the equation [tex]\sqrt{x+5} - \sqrt{x} = 1[/tex] is x = 4.