Problem 10 - 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. $$\frac{8 x}{(4 x-19)(x+2)}$$

Step-by-Step Solution

Answer

The rational expression \(\frac{8x}{(4x-19)(x+2)}\) is undefined for \(x=\frac{19}{4}\) and \(x=-2\).

Step 1: Identify the Denominator

The denominator of the rational function is \((4x-19)(x+2)\). It's crucial to identify the denominator since a rational expression is undefined where the denominator equals zero.

Step 2: Set the Denominator Equal to Zero

Set \((4x-19)(x+2)=0\). This is done because a rational expression is undefined where the denominator is equal to zero.

Step 3: Solve for x

To solve for \(x\), the equation \((4x-19)(x+2) = 0\), is split into two separate equations based on the root theorem. These equations are \(4x-19 = 0\) and \(x+2 = 0\). Solving these gives the values \(x = \frac{19}{4}\) and \(x = -2\)