Solving Polynomial Inequality: A Step-by-Step Guide

Oct 21, 2025 by ADMIN 52 views

Hey guys! Today, we're diving into the world of polynomial inequalities. Specifically, we're going to tackle the inequality $x^2 + 8x + 12 > 0$ . Don't worry, it might look a bit intimidating at first, but we'll break it down step-by-step. We’ll not only solve it but also graph the solution set on a real number line and express our answer in interval notation. So, let's get started!

Understanding Polynomial Inequalities

Before we jump into the nitty-gritty, let's quickly recap what polynomial inequalities are all about. A polynomial inequality is simply an inequality that involves a polynomial expression. Think of it like a regular polynomial equation, but instead of an equals sign (=), we have an inequality sign (>, <, ≥, or ≤). These inequalities help us find the range of values for a variable that satisfy the given condition. When dealing with polynomial inequalities, our main goal is to find the intervals on the number line where the polynomial expression is either positive, negative, or zero, depending on the inequality sign. This involves a series of algebraic manipulations and a bit of logical thinking, but trust me, it's totally doable. Now, with that basic understanding in place, let's dive into our specific example and see how we can solve it step by step!

Step 1: Factor the Polynomial

The first crucial step in solving any polynomial inequality is to factor the polynomial. Factoring helps us identify the critical points where the polynomial might change its sign. In our case, we have $x^2 + 8x + 12 > 0$ . We need to find two numbers that multiply to 12 and add up to 8. Those numbers are 6 and 2. So, we can factor the polynomial as follows:

$x^2 + 8x + 12 = (x + 6)(x + 2)$

Now our inequality looks like this:

$(x + 6)(x + 2) > 0$

Factoring polynomials is a fundamental skill in algebra, and it's incredibly useful for solving various types of equations and inequalities. When we factor a polynomial, we essentially break it down into simpler expressions that are multiplied together. This makes it easier to analyze the behavior of the polynomial, especially when dealing with inequalities. In this specific case, factoring the quadratic expression $x^2 + 8x + 12$ into $(x + 6)(x + 2)$ allows us to identify the roots or zeros of the polynomial, which are the values of $x$ that make the polynomial equal to zero. These roots play a crucial role in determining the intervals where the polynomial is positive or negative, which is essential for solving the inequality.

Step 2: Find the Critical Points

The next step is to find the critical points. These are the values of x that make the polynomial equal to zero. We find them by setting each factor equal to zero and solving for x:

$x + 6 = 0$ => $x = -6$

$x + 2 = 0$ => $x = -2$

So, our critical points are x = -6 and x = -2. These points are super important because they divide the number line into intervals where the polynomial's sign remains consistent. Critical points are like the boundary markers on a number line; they tell us exactly where the polynomial might switch from being positive to negative or vice versa. In the context of inequalities, they are the solutions to the related equation (in our case, $x^2 + 8x + 12 = 0$ ) and serve as the dividing lines for our test intervals. Understanding how to find and interpret critical points is fundamental to solving polynomial inequalities effectively. They help us break down a complex problem into manageable parts, allowing us to determine the intervals where the inequality holds true.

Step 3: Create a Sign Chart

Now, let's create a sign chart. This chart will help us visualize where the polynomial $(x + 6)(x + 2)$ is positive or negative. We'll use our critical points to divide the number line into intervals:

Interval 1: x < -6
Interval 2: -6 < x < -2
Interval 3: x > -2

We’ll pick a test value from each interval and plug it into our factored polynomial to see if the result is positive or negative.

Interval	Test Value	$x + 6$	$x + 2$	$(x + 6)(x + 2)$	Sign
x < -6	-7	-	-	+	+
-6 < x < -2	-3	+	-	-	-
x > -2	0	+	+	+	+

A sign chart is a visual tool that helps us analyze the sign (positive or negative) of a polynomial expression over different intervals on the number line. It's a super organized way to keep track of the sign changes and make sure we don't miss anything. By dividing the number line into intervals based on our critical points, we can systematically test values within each interval to determine the sign of the polynomial expression. The sign chart not only provides a clear picture of where the polynomial is positive or negative but also makes it easier to identify the intervals that satisfy the inequality. It’s like a roadmap that guides us to the correct solution, preventing common mistakes and ensuring accuracy. Creating and interpreting sign charts is a key skill for mastering polynomial inequalities, and once you get the hang of it, you'll find it incredibly helpful.

Step 4: Determine the Solution Set

We want to find where $(x + 6)(x + 2) > 0$ , which means we're looking for intervals where the polynomial is positive. According to our sign chart, this occurs when x < -6 and x > -2.

So, the solution set in interval notation is:

$(-\infty, -6) \cup (-2, \infty)$

Determining the solution set is the ultimate goal when solving any inequality, and it's the point where all our hard work pays off. After creating our sign chart and analyzing the intervals, we can clearly see where the polynomial expression satisfies the given inequality. In this case, we were looking for intervals where $(x + 6)(x + 2) > 0$ , meaning the polynomial is positive. Our sign chart showed us that this occurs when $x < -6$ and $x > -2$ . Translating this information into interval notation allows us to express the solution set in a concise and standard format. The interval notation $(-\infty, -6) \cup (-2, \infty)$ tells us that any value of $x$ less than -6 or greater than -2 will make the inequality true. Understanding how to accurately determine and express the solution set is crucial for solving not just polynomial inequalities but also a wide range of mathematical problems.

Step 5: Graph the Solution Set

Finally, let's graph the solution set on a real number line. We'll mark our critical points -6 and -2. Since the inequality is strict ('>'), we'll use open circles at these points to indicate that they are not included in the solution.

<-------------------|-------------------|-------------------->
      (-\infty)         -6           -2         (\infty)
      o------------------o

We'll shade the intervals where x < -6 and x > -2 to represent our solution set. The graph visually confirms our solution, showing all the values of x that satisfy the inequality.

Graphing the solution set on a real number line is a fantastic way to visualize our solution and make sure it makes sense. The graph provides a clear and intuitive representation of the intervals where the inequality holds true. By marking our critical points and using open or closed circles (depending on whether the endpoints are included), we can easily see the range of values that satisfy the inequality. Shading the appropriate intervals further enhances the visual clarity, making it simple to identify the solution set at a glance. In our case, the graph clearly shows the intervals $(-\infty, -6)$ and $(-2, \infty)$ , confirming that all values of $x$ less than -6 or greater than -2 satisfy the inequality $x^2 + 8x + 12 > 0$ . This graphical representation not only helps us verify our solution but also deepens our understanding of the inequality and its behavior.

Conclusion

And there you have it! We've successfully solved the polynomial inequality $x^2 + 8x + 12 > 0$ , graphed the solution set, and expressed it in interval notation. Remember, the key steps are factoring the polynomial, finding the critical points, creating a sign chart, determining the solution set, and graphing the solution. With practice, you'll become a pro at solving these types of inequalities. Keep up the great work, guys, and happy problem-solving!

Solving polynomial inequalities might seem challenging at first, but by breaking it down into manageable steps, it becomes much more approachable. We've walked through the entire process, from factoring the polynomial to graphing the solution set, and each step plays a crucial role in reaching the final answer. The sign chart is a particularly powerful tool that helps us organize our thoughts and analyze the sign of the polynomial over different intervals. By consistently following these steps and practicing regularly, you'll build confidence and mastery in solving polynomial inequalities. Remember, mathematics is like a muscle – the more you use it, the stronger it gets. So, keep practicing, and you'll be amazed at how quickly you improve! And with that, we've reached the end of our guide. Hopefully, this has clarified any confusion and provided you with the tools to tackle polynomial inequalities with ease. Happy solving, and remember, every problem is an opportunity to learn something new!