Questions: Solve the following equation. Then, state whether the equation is an identity, a conditional equation, or an inconsistent equation. 4/(x-3) + 3/(x+8) = 9/((x+8)(x-3)) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The equation has a single solution. The solution set is -2. (Simplify your answer.) B. The solution set is x x is a real number. C. The solution set is ∅. Is the given equation an identity, a conditional equation, or an inconsistent equation? Identity Conditional equation Inconsistent equation

Solution Steps

We start with the equation: $$\frac{4}{x-3} + \frac{3}{x+8} = \frac{9}{(x+8)(x-3)}$$

The common denominator for the left-hand side is $(x-3)(x+8)$. We rewrite the left-hand side: $$\frac{4(x+8) + 3(x-3)}{(x-3)(x+8)} = \frac{9}{(x+8)(x-3)}$$

Now, we simplify the numerator: $$4(x+8) + 3(x-3) = 4x + 32 + 3x - 9 = 7x + 23$$ Thus, the equation becomes: $$\frac{7x + 23}{(x-3)(x+8)} = \frac{9}{(x+8)(x-3)}$$

Cross-multiplying gives us: $$7x + 23 = 9$$

Now, we solve for $x$: $$7x = 9 - 23$$ $$7x = -14$$ $$x = -2$$

Since we found a single solution $x = -2$, the equation is classified as a conditional equation.

The solution set is: $$\{-2\}$$ And the equation is a conditional equation.

Final Answer