Solving Inequalities: A Step-by-Step Guide
Hey math enthusiasts! Today, we're diving into the world of inequalities. Specifically, we're going to solve the inequality $x^2 + 3x - 18 \geq 0$. This might seem a little daunting at first, but trust me, with a few simple steps, we'll crack this problem and understand the solution like pros. Inequalities are super important in math, and knowing how to solve them is key for understanding a lot of other concepts. So, let's get started, shall we?
This guide will walk you through the process step-by-step. We will cover how to find the critical points, test intervals, and determine the final solution set. This will help you find the $x$ values that satisfy the inequality. Also, we will compare the solution to the multiple-choice options, which will solidify your understanding of this topic. Remember, practice makes perfect! So grab a pen and paper, and let's get those math muscles working! We're not just aiming to find the right answer; we're aiming to truly understand why it's the right answer. We'll explore the underlying concepts, ensuring that you can tackle similar problems with confidence. Let's break down this inequality into manageable chunks, making the learning process both enjoyable and effective. This will give you a solid foundation in solving quadratic inequalities and prepare you for more complex problems down the road. Keep in mind that understanding inequalities is like building a strong foundation for your mathematical journey. Each step is crucial, and each concept builds upon the previous one. We're in this together, and by the end, you'll be able to solve similar problems with confidence.
Step 1: Find the Critical Points
Alright, guys, the first thing we need to do when solving a quadratic inequality like $x^2 + 3x - 18 \geq 0$ is to find the critical points. Critical points are the values of $x$ where the quadratic expression equals zero. Think of them as the spots where the inequality could change direction. To find these points, we need to solve the equation $x^2 + 3x - 18 = 0$. This is a quadratic equation, and we can solve it by factoring, using the quadratic formula, or completing the square. For this problem, factoring is the easiest approach. We're looking for two numbers that multiply to -18 and add up to 3. Those numbers are 6 and -3. So, we can factor the quadratic equation as follows: $(x + 6)(x - 3) = 0$. This factored form gives us our critical points. Setting each factor to zero, we get:
ightarrow x = -6$
ightarrow x = 3$
So, our critical points are $x = -6$ and $x = 3$. These points divide the number line into intervals, which we'll use in the next step to test the inequality. It’s super important to accurately find these critical points, as they serve as the backbone for solving the inequality. Remember, these points represent the values where the quadratic expression equals zero, and they're the boundary lines that separate regions where the inequality holds true or false. Finding these values is like finding the checkpoints in a race, and they define the boundaries of your solution set. Take your time, double-check your factoring or calculations, and ensure you've found the correct values. A mistake here can lead to an incorrect solution! Make sure you grasp the concept of critical points before moving on, as it's a fundamental step in solving any quadratic inequality. With practice, you'll become a pro at finding these critical values and will have a solid understanding of how they work.
Now we've got our critical points, and it's time to move on to the next step. Let's keep the momentum going!
Step 2: Test the Intervals
Okay, now that we have our critical points ($x = -6$ and $x = 3$), we need to test the intervals created by these points. These intervals are:
We need to pick a test value within each interval and substitute it into the original inequality $x^2 + 3x - 18 \geq 0$ to see if it makes the inequality true or false. This process helps us determine which intervals are part of our solution set.
Let's start with the interval $(-\infty, -6)$. We can pick $x = -7$ as our test value. Substituting into the inequality:
This is true! So, the interval $(-\infty, -6)$ is part of our solution.
Next, let's test the interval $(-6, 3)$. We can pick $x = 0$ as our test value. Substituting into the inequality:
This is false! So, the interval $(-6, 3)$ is not part of our solution.
Finally, let's test the interval $(3, \infty)$. We can pick $x = 4$ as our test value. Substituting into the inequality:
This is true! So, the interval $(3, \infty)$ is part of our solution.
We also need to check the critical points themselves. Since the inequality includes