Question

solve the equation.

\\\sqrt{89 - w} - 7 = 3\\

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 the radical, square both sides, and check for extraneous solutions.

Step 1: Isolate the radical term

To get the square root term by itself, add \(7\) to both sides of the equation:

$$\sqrt{89 - w} - 7 = 3$$

$$\sqrt{89 - w} = 3 + 7$$

$$\sqrt{89 - w} = 10$$

Step 2: Eliminate the radical by squaring both sides

Square both sides of the equation to clear the square root:

$$(\sqrt{89 - w})^2 = 10^2$$

$$89 - w = 100$$

Step 3: Solve for \(w\)

Subtract \(89\) from both sides:

$$-w = 100 - 89$$

$$-w = 11$$

Multiply by \(-1\):

$$w = -11$$

Step 4: Verify the solution

Substitute \(w = -11\) back into the original equation to check for extraneous solutions:

$$\sqrt{89 - (-11)} - 7 = 3$$

$$\sqrt{89 + 11} - 7 = 3$$

$$\sqrt{100} - 7 = 3$$

$$10 - 7 = 3$$

$$3 = 3 \quad \text{(True)}$$

The solution is valid.

Answer:

A. The solution set is \(\{-11\}\).