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Chapter 7: Problem 9

Find all numbers for which each rational expression is undefined. If the rational expression is defined for all real numbers, so state. $$\frac{4 x}{(3 x-17)(x+3)}$$

Short Answer

Expert verified
The rational expression \(\frac{4x}{(3x - 17)(x + 3)}\) is undefined for \(x = \frac{17}{3}\) and \(x = -3\)

Step by step solution

01

Set the denominator equal to zero

To find the number value(s) that makes the rational expression undefined, set the entire denominator equal to zero: \((3x - 17)(x + 3) = 0\). This is because dividing by zero is undefined and we need to find which number(s) make the denominator zero.
02

Solve the individual factors

The equation \((3x - 17)(x + 3) = 0\) has two factors. To solve, set each factor equal to 0 separately and solve for x: \[3x - 17 = 0 \rightarrow 3x = 17 \rightarrow x = \frac{17}{3}\] \[x + 3 = 0 \rightarrow x = -3\]
03

Identify the undefined numbers

The equation is undefined when x equals the solutions found above. So, the expression is undefined for \(x = \frac{17}{3}\) and \(x = -3\)

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