We have to solve the expression:
[tex]\sqrt{x+1}-\sqrt{x-2}=5[/tex]We can start by finding the domain of possible values of x.
The argument of the square root can not be negative, so we can limit the values of x as:
[tex]\begin{gathered} 1)\text{ }x+\geq0 \\ x\geq-1 \\ 2)\text{ }x-2\geq0 \\ x\geq2 \end{gathered}[/tex]Then, x has to be greater than 2, as this interval is more restrictive.
When x gets very large, the difference between the two roots approaches 0.
Also, the difference of this square roots decreases with the increase of x, so the maximum value happens for x = 2, which is the minimum value of x in the domain.
We then can calculate the value for the difference of square roots when x = 2 as:
[tex]\begin{gathered} \sqrt{2+1}-\sqrt{2-2} \\ \sqrt{3}-\sqrt{0} \\ \sqrt{3}<5 \end{gathered}[/tex]Then, one side of the equation has a maximum value that is the square root of 3. This maximum value is already smaller than 5, so there is no value of x that can make this expression equal to 5.
Answer: there is no solution for x.
