Q. 3 - Intermediate Algebra

Question

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

$$\begin{gathered} \left(\text{a}\right)\frac{4x^{2}y}{3y} \\ \left(\text{b}\right)\frac{3x-2}{2x+1} \\ \left(\text{c}\right)\frac{u-1}{u^2-3u-28} \end{gathered}$$

Step-by-Step Solution

Answer

$$\begin{gathered} \left(\text{a}\right)\frac{4x^{2}y}{3y}\text{is undefined at} y=0 \\ \left(\text{b}\right)\frac{3x-2}{2x+1}\text{is undefined at} x=-\frac{1}{2} \\ \left(\text{c}\right)\frac{u-1}{u^2-3u-28}\text{is undefined at }u=-4,7. \end{gathered}$$

Step 1 . Given information

$$\begin{gathered} \left(\text{a}\right)\frac{4x^{2}y}{3y} \\ \left(\text{b}\right)\frac{3x-2}{2x+1} \\ \left(\text{c}\right)\frac{u-1}{u^2-3u-28} \end{gathered}$$

Part a . Step 1 . Find the undefined values

$\left(\text{a}\right)\frac{4x^{2}y}{3y}$

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

$$\begin{gathered} 3y=0 \\ \Rightarrow y=0 \end{gathered}$$

Hence, the undefined value at $y=0.$

Part b . Step 1 . Find the undefined values

$\left(\text{b}\right)\frac{3x-2}{2x+1}$

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

$$\begin{gathered} 2x+1=0 \\ \Rightarrow 2x=-1 \\ \Rightarrow x=-\frac{1}{2} \end{gathered}$$

Hence, the undefined value at $x=-\frac{1}{2}.$

Part c . Step 1 . Find the undefined values

$\left(\text{c}\right)\frac{u-1}{u^2-3u-28}$

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

$$\begin{gathered} u^2-3u-28=0 \\ (u-7)(u+4)=0 \\ u-7=0\Rightarrow u=7 \\ u+4=0\Rightarrow u=-4. \end{gathered}$$

Hence, the undefined value at$u=-4,7.$