Question

In the following exercises, determine the values for which the rational expression is defined.

$$\begin{gathered} \left(\text{a}\right)\frac{2x^2}{z} \\ \left(\text{b}\right)\frac{4p-1}{6p-5} \\ \left(\text{c}\right)\frac{n-3}{n^2+2n-8} \end{gathered}$$

Short answer

$$\begin{gathered} \left(\text{a}\right)\frac{2x^2}{z}\text{isundefinedat}z=0 \\ \left(\text{b}\right)\frac{4p-1}{6p-5}\text{isundefinedat}p=\frac{5}{6} \\ \left(\text{c}\right)\frac{n-3}{n^2+2n-8}\text{isundefinedat}n=-4,2 \end{gathered}$$

Step-by-step solution

Step 1. Given information

$$\begin{gathered} \left(\text{a}\right)\frac{2x^2}{z} \\ \left(\text{b}\right)\frac{4p-1}{6p-5} \\ \left(\text{c}\right)\frac{n-3}{n^2+2n-8} \end{gathered}$$

Part a.step 1.find the undefined values

$\left(\text{a}\right)\frac{2x^2}{z}$

Equate the denominator to zero and solve for the unknown variable.

$z=0$

Hence, the undefined value at $z=0.$

Part b.step 1.find the undefined values

$\left(\text{b}\right)\frac{4p-1}{6p-5}$

Equate the denominator to zero and solve for the unknown variable.

$$\begin{gathered} 6p-5=0 \\ 6p=5 \\ p=\frac{5}{6} \end{gathered}$$

Hence, the undefined value at $p=\frac{5}{6}.$

Part c.step 1.find the undefined values

$\left(\text{c}\right)\frac{n-3}{n^2+2n-8}$

Equate the denominator to zero and solve for the unknown variable.

$$\begin{gathered} n^2+2n-8=0 \\ (n+4)(n-2)=0 \\ n+4=0\Rightarrow n=-4 \\ n-2=0\Rightarrow n=2 \end{gathered}$$

Hence, the undefined values are at $n=-4,2.$