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Chapter 7: Problem 10

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state. $$\frac{8 x}{(4 x-19)(x+2)}$$

Short Answer

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

Step by step solution

01

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.
02

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.
03

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 \(x = -2\)

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