Problem 11 - Introductory Algebra for College Students

Question

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state. $$\frac{x+5}{x^{2}+x-12}$$

Step-by-Step Solution

Answer

The given rational expression is undefined for x = 3 and x = -4.

Step 1: Identify the Denominator

Identify the denominator of the rational expression, which is \(x^2 + x - 12\) in this case.

Step 2: Set the Denominator Equal to Zero

The denominator cannot be zero because division by zero is undefined in mathematics. So, to find the values of x that make this expression undefined, set the denominator equal to 0: \(x^2 + x - 12 = 0\)

Step 3: Solve the Equation

Now, you need to solve this equation. First, factor the quadratic: \( (x - 3)(x + 4) = 0 \). Then, set each factor equal to 0 and solve for x: \(x - 3 = 0 => x = 3\) and \(x + 4 = 0 => x = -4 \)