Problem 9 - 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. $$\frac{4 x}{(3 x-17)(x+3)}$$

Step-by-Step Solution

Answer

The rational expression is undefined for \(x = \frac{17}{3}\) and \(x = -3\).

Step 1: Setting Up the Problem

We have a rational expression \( \frac{4x}{(3x-17)(x+3)} \). This can be undefined when the denominator is equal to zero. Therefore, we can start by sounding out both part of the denominator separately, i.e., setting \(3x - 17 = 0\) and \(x + 3 = 0\).

Step 2: Solving the First Part of the Equation

Start with the first part of the denominator, \(3x - 17 = 0\). To solve this equation, add 17 to both sides to isolate the term with x on one side, and then divide by 3 to find the value of x. Like so: \(3x = 17\), which simplifies to \(x = \frac{17}{3}\).

Step 3: Solving the Second Part of the Equation

Next, we look at the second part of the denominator, \(x + 3 = 0\). This one is simpler to solve, just subtract 3 from both sides: \(x = -3\).

Step 4: Final Answer

So, we have \(x = \frac{17}{3}\) and \(x = -3\) as the solutions, which means the given rational expression is undefined at these two points.