Rewrite the given equation without logarithms. Do not solve for .

Solve the equation to find the solution set. Select the correct choice below and, if necessary, fill in the answer
box to complete your choice.
A. The solution set is { }.
(Simplify your answer. Use a comma to separate answers as needed.)
B. There are infinitely many solutions.
C. There is no solution.

Question

Rewrite the given equation without logarithms. Do not solve for

.

Solve the equation to find the solution set. Select the correct choice below and, if necessary, fill in the answer
box to complete your choice.
A. The solution set is { }.
(Simplify your answer. Use a comma to separate answers as needed.)
B. There are infinitely many solutions.
C. There is no solution.

UpStudy AI · Accepted Answer

We start with the given equation:

$$
(x-8)(x+1) = x-24
$$

**Step 1. Rewrite the equation without logarithms**

Since there are no logarithms in the equation, we can directly work with it.

**Step 2. Expand the left-hand side**

Multiply the factors:

$$
(x-8)(x+1) = x^2 + x - 8x - 8 = x^2 - 7x - 8
$$

So the equation becomes:

$$
x^2 - 7x - 8 = x - 24
$$

**Step 3. Rearrange the equation**

Subtract $$x-24$$ from both sides:

$$
x^2 - 7x - 8 - x + 24 = 0
$$

Simplify by combining like terms:

$$
x^2 - 8x + 16 = 0
$$

**Step 4. Factor the quadratic**

Notice that:

$$
x^2 - 8x + 16 = (x-4)^2
$$

Thus the equation becomes:

$$
(x-4)^2 = 0
$$

**Step 5. Determine the solution set**

Since the square equals zero, we have:

$$
x-4 = 0 \quad \Longrightarrow \quad x = 4
$$

**Final Answer:**

The solution set is $$\{4\}$$.
The solution set is {4}.

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