Q. 340 - Intermediate Algebra

Question

In the following exercises, solve.

$\sqrt{x+1}-\sqrt{x-2}=1$

Step-by-Step Solution

Answer

The solution of the given equation is $x=3$.

Step 1. Given information.

The given equation is:

$\sqrt{x+1}-\sqrt{x-2}=1$

Step 2. Simplify the equation.

The given equation can be written as:

$\sqrt{x+1}=1+\sqrt{x-2}$

Take square on both sides.

$\begin{array}{rcl}(\sqrt{x+1}{)}^2& =& (1+\sqrt{x-2}{)}^2\\ x+1& =& 1+2\sqrt{x-2}+(x-2)\\ x+1-1-x+2& =& 2\sqrt{x-2}\\ 2& =& 2\sqrt{x-2}\end{array}$

Step 3. Solve the equation.

Divide both sides by 2.

$$\begin{gathered} \frac{2}{2}=\frac{2\sqrt{x-2}}{2} \\ 1=\sqrt{x-2} \end{gathered}$$

Square on both sides.

$\begin{array}{rcl}{1}^2& =& (\sqrt{x-2}{)}^2\\ 1& =& x-2\\ 1+2& =& x\\ 3& =& x\end{array}$

Step 4. Conclusion.

The solution of the given equation is $x=3$.