Solving Inequalities: A Step-by-Step Guide With Interval Notation

Hey math enthusiasts! Let's dive into the world of inequalities and learn how to solve them, specifically focusing on the example: $\frac{6-x}{x+2} \geq 0$. We'll break it down step-by-step, making sure you grasp the concepts and can confidently tackle similar problems. The goal here is not just to find the answer but to understand why the answer is what it is. This knowledge will be super helpful as you progress in your math journey. We'll also cover how to express the solution in interval notation, which is a concise way to represent sets of numbers. So, grab your pencils, and let's get started!

Understanding the Basics: Inequality Concepts

Before we jump into solving the specific inequality, let's refresh our understanding of inequalities in general. Inequalities are mathematical statements that compare two expressions using symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Unlike equations, which state that two expressions are equal, inequalities tell us about the relative size of the expressions. For our problem, $\frac{6-x}{x+2} \geq 0$, we are looking for values of x that make the expression $\frac{6-x}{x+2}$ greater than or equal to zero. This means we're looking for where the expression is positive or zero. This simple concept is foundational for understanding the solution. The key to solving inequalities lies in identifying the critical points, where the expression changes sign (from positive to negative or vice versa). These critical points are the values of x that make the numerator or denominator equal to zero (or undefined).

When working with inequalities, there are a few important rules to keep in mind. One crucial rule is that when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol. This is because multiplying or dividing by a negative number flips the order of the numbers on the number line. For instance, if you have 2 < 4 and multiply both sides by -1, you get -2 > -4. This rule is essential to remember because it can lead to mistakes if overlooked. Additionally, we must be careful when dealing with fractions and expressions involving variables in the denominator. A denominator cannot be zero because division by zero is undefined. This restriction is crucial, and it means we have to exclude any values of x that would make the denominator zero from our solution set. Now, let's see how this all applies to our example.

To summarize, inequalities are about comparing the size of expressions, critical points are key to finding solutions, and remembering the rules about multiplying/dividing by negative numbers and avoiding division by zero is crucial. Ready to move on? Let's solve our inequality.

Step-by-Step Solution: Unraveling the Inequality

Alright, let's roll up our sleeves and solve the inequality $\frac{6-x}{x+2} \geq 0$. We will break down the process into easy-to-follow steps. First, we need to find the critical points. These are the values of x that make the numerator or the denominator equal to zero. To find the critical points from the numerator, we set the numerator equal to zero:

• 6 - x = 0
• x = 6

So, x = 6 is one of our critical points. Next, to find the critical points from the denominator, we set the denominator equal to zero:

• x + 2 = 0
• x = -2

Therefore, x = -2 is our second critical point. Note that x = -2 is not included in the solution because it makes the denominator zero (undefined). Having found our critical points, we now create a number line and mark these points on it. This number line will help us visualize the intervals where the expression is positive, negative, or zero. Since -2 is not included in the solution (because it makes the expression undefined), we use an open circle at -2. However, 6 is included in the solution (since it makes the expression equal to zero), so we'll use a closed circle at 6. The critical points divide the number line into three intervals: (-∞, -2), (-2, 6], and [6, ∞). This is a crucial step! Now, we test each interval to determine whether the expression $\frac{6-x}{x+2}$ is positive or negative within each interval. To do this, we choose a test value within each interval and substitute it into the expression. This will show whether the expression is positive or negative in that interval.

For the interval (-∞, -2), let's choose x = -3:

• $\frac{6 - (-3)}{-3 + 2} = \frac{9}{-1} = -9$. Since the result is negative, the expression is negative in the interval (-∞, -2).

For the interval (-2, 6], let's choose x = 0:

• $\frac{6 - 0}{0 + 2} = \frac{6}{2} = 3$. Since the result is positive, the expression is positive in the interval (-2, 6].

For the interval [6, ∞), let's choose x = 7:

• $\frac{6 - 7}{7 + 2} = \frac{-1}{9} = -\frac{1}{9}$. Since the result is negative, the expression is negative in the interval [6, ∞).

We are looking for where the expression is greater than or equal to zero (positive or zero). Therefore, the solution includes the interval (-2, 6] because the expression is positive in this interval, and also includes the value 6 because the expression equals 0 at x = 6. The solution excludes -2 because the expression is undefined there. Ready to express this solution in interval notation?

Expressing the Solution in Interval Notation

We have determined that the solution to the inequality $\frac{6-x}{x+2} \geq 0$ includes the interval (-2, 6], which means x can be any value greater than -2 and less than or equal to 6. But what exactly is interval notation, and how do we write our solution using it? Interval notation is a way to represent a set of numbers using parentheses and brackets. Parentheses ( ) are used to indicate that the endpoint is not included in the interval, and brackets [ ] are used to indicate that the endpoint is included. When using interval notation, we always write the smaller number first and the larger number second. For the solution to our inequality, the interval starts at -2 (but does not include -2, because the denominator would be 0, making the expression undefined) and goes up to and includes 6. This is expressed in interval notation as follows:

• (-2, 6]

The parenthesis next to -2 indicates that -2 is not included in the solution, and the bracket next to 6 indicates that 6 is included in the solution. This is a compact and clear way to represent the set of all x values that satisfy the inequality. The solution set in interval notation is (-2, 6]. This means all real numbers greater than -2 and less than or equal to 6. You are doing great!

To make sure we understand this, let's recap. We started with the inequality, found the critical points, used a number line to organize our intervals, tested those intervals, and now we've written our solution in interval notation. This process is key when solving inequalities. Using interval notation gives you a clear and concise way to write down your answers, making it easy to understand the solution set. It's a fundamental concept in mathematics that you'll use time and again. The ability to switch between inequality notation, number lines, and interval notation will become second nature with practice.

Conclusion: Mastering Inequalities

And there you have it! We've successfully solved the inequality $\frac{6-x}{x+2} \geq 0$ and expressed the solution in interval notation. Let's recap what we've learned. First, we understood what inequalities are and how they differ from equations. We then identified the critical points by finding the zeros of the numerator and denominator. Next, we used the critical points to divide the number line into intervals, testing each interval to see whether the expression was positive or negative. Finally, we expressed our solution, remembering to use parentheses and brackets correctly. Remember that an open circle means the endpoint is not included in the solution, and a closed circle indicates it is included. The answer is (-2, 6].

Solving inequalities is a critical skill in algebra and beyond. It's used in all sorts of problems in calculus, statistics, and engineering. The method we used here, involving critical points and testing intervals, can be applied to many other types of inequalities. Keep practicing these types of problems, and you'll become more and more comfortable with the process. Try solving other inequalities on your own. Remember to always check your answers to make sure they make sense. Keep up the excellent work! You're doing great, and keep exploring! Congratulations on making it through this guide!"