Question

solve the equation.

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

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the solution set is \\{\quad\\}.
(simplify your answer. type an integer or a fraction. use a comma to separate answers as needed.)

b. the solution set is the empty set.

Explanation:

🆕 New Concept Discovered: Solving Radical Equations
Isolate a radical and square both sides to eliminate roots.

Step 1: Isolate one radical term

To make squaring easier, move one of the radical terms to the other side of the equation.

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

Step 2: Square both sides

Square both sides of the equation to eliminate the outer radical on the left.

$$ (\sqrt{x + 2})^2 = (1 + \sqrt{x - 3})^2 $$

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

Step 3: Simplify and isolate the remaining radical

Combine like terms on the right side and isolate the term containing the radical.

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

Subtract \( x \) from both sides:

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

Add \( 2 \) to both sides:

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

Divide both sides by \( 2 \):

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

Step 4: Square both sides again

Square both sides to eliminate the remaining radical.

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

$$ 4 = x - 3 $$

$$ x = 7 $$

Step 5: Verify the solution

Check if \( x = 7 \) satisfies the original equation to ensure it is not an extraneous solution.

$$ \sqrt{7 + 2} - \sqrt{7 - 3} = \sqrt{9} - \sqrt{4} $$

$$ 3 - 2 = 1 $$

Since \( 1 = 1 \), the solution is valid.

Answer:

A. The solution set is \(\{7\}\).