Problem 103 - Algebra and Trigonometry

Question

List all numbers that must be excluded from the domain of each expression. $$\frac{|x-1|-3}{|x+2|-14}$$

Step-by-Step Solution

Answer

The numbers that must be excluded from the domain of the expression \( \frac{|x-1|-3}{|x+2|-14} \) are 12 and -16.

Step 1: Set the Denominator Equal to Zero

For the expression \( \frac{|x-1|-3}{|x+2|-14} \), determine where the denominator equals zero. Therefore, take the denominator \( |x+2| - 14 \) and set it equal to zero, resulting in the equation \( |x+2| - 14 = 0 \).

Step 2: Solve the Equation

For the equation \( |x+2| - 14 = 0 \), rearrange the equation to isolate the absolute value term to one side, resulting in \( |x+2| = 14 \). When the absolute value of a quantity is set equal to a positive number, we consider both the positive and negative version of that number. Therefore we have, \( x+2 = 14 \) and \( x+2 = -14 \).

Step 3: Solve for x

Solving for \( x \) in both equations \( x+2 = 14 \) and \( x+2 = -14 \) gives us \( x = 14 - 2 = 12 \) and \( x = -14 - 2 = -16 \). Therefore, the numbers that must be excluded from the domain of the expression \( \frac{|x-1|-3}{|x+2|-14} \) are 12 and -16, because at these x values, the expression is undefined.