Question

solve the radical equation.
\sqrt{2x + 3} + \sqrt{x - 2} = 2

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\bigcirc a. the solution set is \\{\\}
(simplify your answer. use a comma to separate answers as needed.)
\bigcirc b. the solution set is the empty set.

Explanation:

Step1: Define domain of variables

For radicals to be real:
$2x+3 \geq 0 \implies x \geq -\frac{3}{2}$
$x-2 \geq 0 \implies x \geq 2$
Combined domain: $x \geq 2$

Step2: Isolate one radical

$\sqrt{2x+3} = 2 - \sqrt{x-2}$

Step3: Square both sides

$(\sqrt{2x+3})^2 = (2 - \sqrt{x-2})^2$
$2x+3 = 4 - 4\sqrt{x-2} + x - 2$

Step4: Simplify the equation

$2x+3 = x + 2 - 4\sqrt{x-2}$
$x + 1 = -4\sqrt{x-2}$

Step5: Analyze equation validity

Left side: $x \geq 2 \implies x+1 \geq 3 > 0$
Right side: $-4\sqrt{x-2} \leq 0$
Positive value cannot equal non-positive value.

Step6: Verify no solutions exist

No $x \geq 2$ satisfies the equation.

Answer:

B. The solution set is the empty set.