Question

$\sqrt{2x + 5} + \sqrt{x - 6} = 3$
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

First, find valid $x$ values:
For $\sqrt{2x+5}$: $2x+5\geq0 \implies x\geq-\frac{5}{2}$
For $\sqrt{x-6}$: $x-6\geq0 \implies x\geq6$
Combined domain: $x\geq6$

Step2: Isolate one radical

Rearrange the equation:
$\sqrt{2x+5}=3-\sqrt{x-6}$

Step3: Square both sides

Eliminate the left radical:
$(\sqrt{2x+5})^2=(3-\sqrt{x-6})^2$
$2x+5=9-6\sqrt{x-6}+(x-6)$

Step4: Simplify the equation

Combine like terms:
$2x+5=x+3-6\sqrt{x-6}$
$x+2=-6\sqrt{x-6}$

Step5: Analyze equation validity

Left side: For $x\geq6$, $x+2\geq8>0$
Right side: $-6\sqrt{x-6}\leq0$
A positive number cannot equal a non-positive number.

Answer:

B. The solution set is the empty set.