Problem 91 - Algebra and Trigonometry

Question

Solve each inequality in Exercises \(86-91\) using a graphing utility. $$ \frac{1}{x+1} \leq \frac{2}{x+4} $$

Step-by-Step Solution

Answer

The x-coordinate values on which the plotted function intersects or is below the line \( y = 0 \) forms the solution to the inequality.

Step 1: Rewrite the inequality

In order to solve the inequality using a graphing utility, it's easier to have all terms on one side of the inequality. Hence the inequality becomes \( \frac{1}{x+1} - \frac{2}{x+4} \leq 0 \)

Step 2: Plot the inequality

Now, plot \( y = \frac{1}{x+1} - \frac{2}{x+4} \) using a graphing utility. The line \( y = 0 \) is also plotted for visual reference as the inequality is set to be less than or equal to zero.

Step 3: Determine the solution from the graph

From the graph, identify where the plotted function is below or intersect the line \( y = 0 \). This x-coordinate values are the solution to the inequality. Be careful about endpoints of the function, which might occur where the function is undefined due to division by zero.