Problem 97 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I began the solution of the rational inequality \(\frac{x+1}{x+3} \geq 2\) by setting both \(x+1\) and \(x+3\) equal to zero.

Step-by-Step Solution

Answer

The statement does not make sense due to the incorrect approach of solving the inequality. The right approach is to isolate the rational expression on one side of the inequality, then to find the critical points where the inequality is undefined or zero. Setting the numerator and the denominator to zero at the same time doesn't make sense. The correct critical points are when \(x = -1\) and \(x = -3\), separately.

Step 1: Evaluate the statement

The statement claims to start the solution for the inequality \(\frac{x+1}{x+3} \geq 2\) by setting both \(x+1\) and \(x+3\) equal to zero. This claim doesn't make sense because to solve this inequality, the correct starting point is to isolate the rational expression on one side of the inequality sign, which in this case is already satisfied.

Step 2: Find the correct starting point for solving the inequality

To solve this inequality, the right approach is to get it in the correct form and then find the critical points where the inequality is undefined or zero. Here the correct form means to have 0 on one side of the inequality.

Step 3: Explain the correct methods

Next, subtract 2 from both sides to rearrange the inequality into the correct form: \(\frac{x+1}{x+3}-2\geq 0\). After this step, the critical points can be found.

Step 4: Demonstrating the correct methods

To simplify, find a common denominator and combine terms. Then, solve the equation in the numerator and the denominator separately to find the critical points. The correct critical points are the solutions to \(x + 1 = 0\) and \(x+3 = 0\), separately. So, \(x = -1\) and \(x = -3\) are the critical points, not \(x = -1\) and \(x = -3\) simultaneously.