Solving Inequalities: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into the world of inequalities. Today, we're going to tackle a specific problem: $\frac{2x}{x+4} - \frac{1}{x+4} < 0$. We'll not only solve this inequality but also visualize the solution through a graph and express it in the clear and concise language of interval notation. Ready to break it down? Let's get started!

Understanding the Basics: Inequalities and Their Solutions

First things first, what exactly is an inequality? Well, it's a mathematical statement that compares two values, showing that they are not equal. Instead of an equals sign (=), inequalities use symbols like these: less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥). Solving an inequality means finding all the values of the variable (in our case, 'x') that make the statement true. The solution to an inequality isn't usually a single number, but rather a range of numbers. This is where interval notation and graphs come in handy; they help us represent and understand these ranges.

The Critical Role of Interval Notation

Interval notation is a handy way to write the set of all possible solutions to an inequality. It uses parentheses ( ) and brackets [ ] to indicate whether the endpoints of the interval are included or excluded. A parenthesis means the endpoint is not included (think of it as an open circle on a graph), while a bracket means the endpoint is included (a closed circle on a graph). For instance, if our solution is all numbers greater than 2, we write this in interval notation as (2, ∞). The parenthesis next to '2' shows that '2' itself isn't part of the solution. The infinity symbol (∞) always gets a parenthesis because infinity isn't a specific number we can include.

Graphing the Solution: A Visual Aid

Graphs provide a visual representation of the solution to an inequality. We typically use a number line to plot the solution. For example, if our solution is x > 2, we would draw an open circle at '2' on the number line and shade the line to the right, showing that all values greater than 2 are included. If the solution were x ≥ 2, we'd use a closed circle at '2' to show that '2' itself is part of the solution. The visual nature of a graph makes it easier to understand the range of values that satisfy the inequality.

Solving the Inequality: $\frac{2x}{x+4} - \frac{1}{x+4} < 0$

Now, let's get down to business and solve our inequality: $\frac{2x}{x+4} - \frac{1}{x+4} < 0$. Here's a step-by-step breakdown to help you follow along:

Step 1: Simplify the Inequality

Since the two fractions on the left side of the inequality have the same denominator (x + 4), we can combine them into a single fraction: $\frac{2x - 1}{x + 4} < 0$. This simplifies our problem and makes it easier to manage.

Step 2: Find Critical Points

Critical points are the values of 'x' where the expression on the left side either equals zero or is undefined. To find these points, we set both the numerator and the denominator equal to zero and solve for x.

• Numerator: $2x - 1 = 0$. Solving for x, we get $x = \frac{1}{2}$.
• Denominator: $x + 4 = 0$. Solving for x, we get $x = -4$. Note that x = -4 is not in the domain of the function, because the denominator cannot be zero. Therefore, $x=-4$ is an undefined point.

These critical points, $x = \frac{1}{2}$ and $x = -4$, divide the number line into intervals. Our solutions will lie within these intervals.

Step 3: Create a Sign Chart

A sign chart helps us determine the sign (positive or negative) of the expression $\frac{2x - 1}{x + 4}$ within each interval created by the critical points. We set up the chart like this:

We select a test value within each interval and substitute it into the numerator ($2x - 1$) and the denominator ($x + 4$) to see if the resulting value is positive or negative. Then, we divide the sign of the numerator by the sign of the denominator to determine the overall sign of the fraction. The sign chart simplifies the process.

Step 4: Determine the Solution

We want to find where $\frac{2x - 1}{x + 4} < 0$, which means we're looking for the intervals where the expression is negative. From our sign chart, we see that the expression is negative in the interval $(-4, \frac{1}{2})$.

Step 5: Write the Solution in Interval Notation

Therefore, the solution to the inequality $\frac{2x}{x+4} - \frac{1}{x+4} < 0$ is $(-4, \frac{1}{2})$. The parentheses indicate that the endpoints -4 and $\frac{1}{2}$ are not included in the solution.

Graphing the Solution

To graph the solution, draw a number line. Mark -4 and $\frac{1}{2}$ on the number line. Since our interval is $(-4, \frac{1}{2})$, we'll use an open circle at both -4 and $\frac{1}{2}$ to show that these points are not included. Then, shade the region between -4 and $\frac{1}{2}$ to represent all the values that satisfy the inequality. This visual representation clarifies which values of 'x' make the inequality true.

Key Concepts and Recap

• Inequality: A mathematical statement comparing two values that are not equal. We use symbols like <, >, ≤, and ≥.
• Interval Notation: A concise way to express the set of all possible solutions to an inequality, using parentheses ( ) and brackets [ ].
• Critical Points: Values where the expression equals zero or is undefined. They are crucial for creating sign charts.
• Sign Chart: A tool to determine the sign of the expression in different intervals created by critical points.

Frequently Asked Questions (FAQ)

Why do we use open circles at -4 and 1/2 in the graph?

Because the inequality uses the “less than” symbol (<), not “less than or equal to” (≤). This means the critical points themselves are not part of the solution.

What happens if the inequality were $\frac{2x}{x+4} - \frac{1}{x+4} \le 0$?

If the inequality were ≤, we would include the value where the expression equals zero. In our example, the solution would be $(-4, \frac{1}{2}]$, with a bracket at $\frac{1}{2}$ to show it is included. However, note that -4 is always excluded due to the denominator.

How important is the sign chart?

The sign chart is super important! It's the most reliable way to find the intervals where the inequality holds true. It helps organize your work and minimizes the chances of making mistakes.

Conclusion: Mastering Inequalities

And there you have it, folks! We've successfully solved an inequality, graphed its solution, and expressed it in interval notation. Remember, practice is key. The more you work through these problems, the more comfortable you'll become. Inequalities are a fundamental concept in mathematics and have applications in many areas. Keep practicing, and you'll be acing these problems in no time! Remember to always check your answers and ensure your solution makes sense within the context of the problem.