Problem 103 - College Algebra Essentials

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 given expression are 12 and -16.

Step 1: Understand the Problem

The expression given is a ratio of two expressions: the numerator is \(|x-1|-3\) and the denominator is \(|x+2|-14\). A rational expression is undefined when its denominator equals zero. To find the values for which the expression is undefined, we need to equate the denominator to zero and solve for \(x\). In this case, we need to solve the equation \(|x+2|-14 = 0\).

Step 2: Solve the Equation

First, isolate the absolute value on one side of the equation. Add 14 to both sides to get \(|x+2| = 14\). Now we know that the quantity inside the absolute value, \(x+2\), equals either 14 or -14. So, we make two separate equations from this absolute equation: \(x+2 = 14\) and \(x+2 = -14\). Solving these equations, we get \(x = 12\) and \(x = -16\).

Step 3: Conclusion

The two numbers that must be excluded from the domain of the given expression are 12 and -16, because they make the denominator equal to zero, and an expression is undefined when its denominator equals zero.