Problem 91 - College Algebra

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 solution for the inequality \(\frac{1}{x+1} \leq \frac{2}{x+4}\) can be obtained by graphing the functions and observing where \(y_1\) is less than or equal to \(y_2\). This will provide the x-coordinates (or intervals) that satisfy the inequality.

Step 1: Graphing the functions

Plot the two functions \(y_1 = \frac{1}{x+1}\) and \(y_2 = \frac{2}{x+4}\) using a graphing utility. Observe where they intersect. This intersection point represents the value of x for which both functions are equal.

Step 2: Interpret the graph

Now, for the portion of the graph where \(y_1\) is less than or equal to \(y_2\), the x-coordinates give the solution for the inequality which is \(\frac{1}{x+1} \leq \frac{2}{x+4}\).

Step 3: Determining the interval

The interval (or intervals) that includes all the x-coordinates that satisfy the initial inequality represents the solution of the inequality.

Key Concepts

Graphing Functions

Graphing functions is a powerful way to visualize mathematical relationships. In this exercise, we're comparing two rational functions:

• \( y_1 = \frac{1}{x+1} \)
• \( y_2 = \frac{2}{x+4} \)

These are equations involving variables in the denominator. Using a graphing utility, such as a graphing calculator or software, helps us draw these curves and understand their behavior.
When plotting these functions, observe if they have asymptotes, points where the function approaches infinity and never touches a certain line, often occurring at values that make the denominator zero.
For instance, \( y_1 \) has an asymptote at \( x = -1 \), while \( y_2 \) has one at \( x = -4 \). Understanding these characteristics can help you accurately sketch and analyze the graphs.

Intersection Points

Intersection points are where two graphs meet or cross over one another. This tells us where the outputs of the functions are the same for a given input.
When graphing functions like our rational functions, this is crucial. To find these points:

• Use your graphing utility to plot both functions on the same coordinate plane.
• Look for points where the curves of \( y_1 \) and \( y_2 \) intersect.
• This intersection corresponds to solutions for the equation \( \frac{1}{x+1} = \frac{2}{x+4} \).

Finding the intersection helps simplify the process of solving inequalities by allowing us to see the relationship between the equations visually. It tells us exactly where the functions are equal, which is critical for understanding solution intervals.

Solution Intervals

Solution intervals represent the range of \( x \)-values that satisfy a given inequality. In the context of our exercise, we are looking for where \( \frac{1}{x+1} \leq \frac{2}{x+4} \).
To find these intervals, examine the graph:

• Once the functions are graphed, look for the sections where the graph of \( y_1 \) lies below or touches \( y_2 \).
• These overlapping regions on the \( x \)-axis indicate where the inequality holds true.
• Identify the critical points, such as intersection points and asymptotes, to help define the start and end of these intervals.

So, by analyzing these regions, you can determine the \( x \)-values that make the initial inequality true, and effectively understand the problem's solution set.