Question

solve the equation.
\\\sqrt{x + 7} = \sqrt{2x + 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. use a comma to separate answers as needed.)
b. the solution set is the empty set.

Explanation:

⚡ Using what you learned: Roots and Radicals

Step 1: Square both sides

To eliminate the square roots, square both sides of the equation:

$$ (\sqrt{x + 7})^2 = (\sqrt{2x + 1})^2 $$

$$ x + 7 = 2x + 1 $$

Step 2: Solve for x

Subtract \( x \) from both sides:

$$ 7 = x + 1 $$

Subtract \( 1 \) from both sides:

$$ x = 6 $$

Step 3: Verify the solution

Substitute \( x = 6 \) back into the original equation to check for extraneous solutions:

$$ \sqrt{6 + 7} = \sqrt{2(6) + 1} $$

$$ \sqrt{13} = \sqrt{12 + 1} $$

$$ \sqrt{13} = \sqrt{13} $$

The solution is valid.

Answer:

A. The solution set is {6}.