Solving Quadratic Inequalities: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of quadratic inequalities. Specifically, we're going to tackle the inequality $x^2 + 3x - 28 < 0$. Inequalities might seem a little intimidating at first, but don't worry, we'll break it down step-by-step so you can master them. By the end of this guide, you'll not only know how to solve this particular inequality but also understand the general approach for tackling any quadratic inequality that comes your way. So, let's put on our math hats and get started!

Understanding Quadratic Inequalities

First off, let's clarify what we're actually dealing with. A quadratic inequality is an inequality that involves a quadratic expression (that is, an expression with a variable raised to the power of 2). The general form looks something like $ax^2 + bx + c < 0$, $ax^2 + bx + c > 0$, $ax^2 + bx + c \leq 0$, or $ax^2 + bx + c \geq 0$, where 'a', 'b', and 'c' are constants, and 'x' is our variable. The key difference between a quadratic equation and a quadratic inequality is that instead of looking for specific values of 'x' that make the expression equal to zero, we're looking for a range of values that make the expression either less than zero, greater than zero, less than or equal to zero, or greater than or equal to zero. This means our solutions will often be intervals rather than single numbers. To get a good grasp, it's crucial to familiarize yourself with the properties of quadratic expressions. This includes understanding the shape of a parabola (the graph of a quadratic function), how the sign of the leading coefficient ('a') affects whether the parabola opens upwards or downwards, and the significance of the roots (x-intercepts) of the quadratic equation. The roots, in particular, play a pivotal role in determining the intervals where the quadratic expression is positive or negative, which is exactly what we need to solve inequalities.

Step 1: Factor the Quadratic Expression

The heart of solving a quadratic inequality lies in factoring the quadratic expression. Factoring transforms the expression into a product of two linear factors, which makes it much easier to analyze its sign. In our case, we have $x^2 + 3x - 28$. We need to find two numbers that multiply to -28 and add to 3. After a little thought, we can see that these numbers are 7 and -4. So, we can factor the expression as follows:

$x^2 + 3x - 28 = (x + 7)(x - 4)$

So, our inequality now becomes:

$(x + 7)(x - 4) < 0$

Factoring correctly is super important! Double-check your factorization by expanding the factored form to make sure it matches the original quadratic expression. If you're struggling with factoring, there are tons of resources available online, including video tutorials and practice problems. Mastering factoring is not just essential for solving quadratic inequalities, but also for many other areas of algebra and calculus. Moreover, don't hesitate to use online calculators or tools to verify your factorization, especially when dealing with more complex expressions. However, the goal should always be to understand the process and be able to factor by hand, as this will give you a much deeper understanding of the underlying concepts.

Step 2: Find the Critical Points

The next step is to find what we call the critical points. These are the values of 'x' that make the expression equal to zero. In other words, they are the roots of the quadratic equation $(x + 7)(x - 4) = 0$. To find them, we simply set each factor equal to zero and solve for 'x':

• $x + 7 = 0 \Rightarrow x = -7$
• $x - 4 = 0 \Rightarrow x = 4$

So, our critical points are x = -7 and x = 4. These critical points are crucial because they divide the number line into intervals. The sign of the expression $(x + 7)(x - 4)$ will be constant within each interval. This is because the sign can only change at the points where the expression equals zero (the critical points). Think of these critical points as the potential turning points for the expression – the places where it might switch from being positive to negative, or vice versa. By identifying these points, we're essentially mapping out the landscape of the expression's behavior. Understanding this concept is vital for correctly solving the inequality, as it allows us to test intervals rather than every single value of 'x'.

Step 3: Create a Sign Chart

Now, this is where things get visual! We're going to create a sign chart. This chart helps us determine the sign of the expression $(x + 7)(x - 4)$ in each of the intervals created by our critical points. Draw a number line and mark the critical points, -7 and 4, on it. These points divide the number line into three intervals: $(-\infty, -7)$, $(-7, 4)$, and $(4, \infty)$.

    <----------------|----------------|---------------->
        (-∞, -7)         (-7, 4)          (4, ∞)
                 -7                4

Now, we need to pick a test value within each interval and plug it into the factored expression $(x + 7)(x - 4)$ to see if the result is positive or negative.

• Interval $(-\infty, -7)$: Let's pick x = -8.
  • $(x + 7) = (-8 + 7) = -1$ (negative)
  • $(x - 4) = (-8 - 4) = -12$ (negative)
  • $(x + 7)(x - 4) = (-1)(-12) = 12$ (positive)
• Interval $(-7, 4)$: Let's pick x = 0.
  • $(x + 7) = (0 + 7) = 7$ (positive)
  • $(x - 4) = (0 - 4) = -4$ (negative)
  • $(x + 7)(x - 4) = (7)(-4) = -28$ (negative)
• Interval $(4, \infty)$: Let's pick x = 5.
  • $(x + 7) = (5 + 7) = 12$ (positive)
  • $(x - 4) = (5 - 4) = 1$ (positive)
  • $(x + 7)(x - 4) = (12)(1) = 12$ (positive)

We can now complete our sign chart:

    <--------(+)-------|--------(-)-------|--------(+)------->
        (-∞, -7)         (-7, 4)          (4, ∞)
                 -7                4

The sign chart is your visual guide to the solution. It shows you exactly where the quadratic expression is positive, negative, or zero. Understanding how to construct and interpret a sign chart is a fundamental skill in solving inequalities.

Step 4: Determine the Solution

Remember, we want to find the values of 'x' for which $(x + 7)(x - 4) < 0$. This means we're looking for the intervals where the expression is negative. Looking at our sign chart, we see that the expression is negative in the interval $(-7, 4)$. Since our inequality is strictly less than zero (and not less than or equal to zero), we don't include the endpoints -7 and 4 in our solution. Therefore, the solution to the inequality is:

$-7 < x < 4$

This is often written in interval notation as $(-7, 4)$. So, the sign chart is the key. It visually represents the solution to the inequality. Make sure you understand how to read it!

Conclusion

And there you have it! We've successfully solved the quadratic inequality $x^2 + 3x - 28 < 0$. The solution is $-7 < x < 4$. Remember, the key steps are: factoring the quadratic expression, finding the critical points, creating a sign chart, and then using the sign chart to determine the intervals that satisfy the inequality. Solving quadratic inequalities might seem tricky at first, but with practice, you'll become a pro in no time. Don't be afraid to work through more examples and try different types of inequalities. The more you practice, the more confident you'll become. So, keep practicing, and you'll master this skill in no time! Good luck, and happy problem-solving!