Problem 8 - 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 \( x = 17/3 \) and \( x = -3 \).

Step 1: Identify the denominator

The denominator of the rational function is \( (3x - 17)(x + 3) \). The goal is to set this equal to zero and solve for \( x \), because the function is undefined at these values.

Step 2: Set each factor of the denominator equal to zero

It is important to remember that if a product of factors equals zero, then at least one of the factors must be zero. Therefore, set each factor in the denominator equal to zero to solve for \( x \):\n\n\( 3x - 17 = 0 \)\n\nand \n\n\( x + 3 = 0 \)

Step 3: Solve each equation

Solving each equation gives the value(s) of \( x \) for which the denominator and thus the function are undefined. \n\nFor \( 3x - 17 = 0 \), add 17 to both sides of the equation and then divide by 3 to solve for \( x \), which gives \( x = 17/3 \). \n\nFor \( x + 3 = 0 \), subtract 3 from both sides of the equation to solve for \( x \), which gives \( x = -3 \).