Problem 12 - 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. $$\frac{7 x-14}{x^{2}-9 x+20}$$

Step-by-Step Solution

Answer

The rational expression is undefined for \(x=4\) and \(x=5\).

Step 1: Identify the denominator of the rational expression

The denominator of the given rational expression is \(x^{2}-9 x+20\). With this expression, we can now start figuring out the values of \(x\) that make the rational expression undefined.

Step 2: Set the denominator equal to zero, and solve for \(x\)

The rational expression is undefined where the denominator equals zero. Therefore, to find out when the rational expression is undefined, set the denominator equal to zero: \[x^{2}-9 x+20=0\] This equation is a simple quadratic equation, which can be solved by a number of methods. Factoring is simple in this case: \[(x - 4)(x - 5) = 0\] Therefore, the solutions to this are \(x=4\) and \(x=5\).

Step 3: Identify the undefined points

Thus, the given rational expression is undefined for \(x=4\) and \(x=5\).