Hey guys! Today, we're diving into the world of quadratic inequalities. Specifically, we're going to tackle the inequality x^2 - 4x - 5 ≤ 0. Our goal? To express the solution in that neat little interval notation. So, buckle up and let's get started!
Understanding Quadratic Inequalities
Before we jump into solving, let’s chat a bit about what quadratic inequalities are all about. You see, a quadratic inequality is just like a regular quadratic equation, but instead of an equals sign (=), we've got an inequality sign (like <, >, ≤, or ≥). These inequalities help us find ranges of values rather than specific solutions, and that's where things get interesting.
So, why are we even bothering with this stuff? Well, quadratic inequalities pop up in all sorts of places, from physics to economics. They help us model situations where we're not just looking for a single answer, but rather a whole bunch of possibilities. Think about scenarios where you want to know when a projectile's height is below a certain level or when a company's profits are above a certain target. That's where these inequalities shine!
Now, when we're dealing with quadratic inequalities, the name of the game is to figure out the range of x-values that make the inequality true. This isn't as straightforward as solving a regular equation. We're not just looking for a couple of points; we're looking for intervals – chunks of the number line where the inequality holds water. To get there, we’re going to use a mix of factoring, finding critical points, and testing intervals. Trust me; it’s not as scary as it sounds!
Step-by-Step Solution to x^2 - 4x - 5 ≤ 0
Okay, let’s break down the solution to our inequality, x^2 - 4x - 5 ≤ 0, step by step. I promise, it's like following a recipe – just stick to the steps, and you'll get there!
Step 1: Factor the Quadratic Expression
First things first, we need to factor the quadratic expression. Factoring helps us find the roots, which are super important for solving inequalities. Our expression is x^2 - 4x - 5. We're looking for two numbers that multiply to -5 and add up to -4. Those numbers are -5 and 1. So, we can factor the expression like this:
(x - 5)(x + 1) ≤ 0
See how we broke it down? Factoring is like reverse engineering the multiplication. Once we have the factored form, we're one step closer to cracking the code.
Step 2: Find the Critical Points
The critical points are the values of x that make the expression equal to zero. They are also called roots or zeros of the quadratic equation. These points are crucial because they divide the number line into intervals, and the inequality's sign might change at these points. So, let's set each factor equal to zero and solve for x:
- x - 5 = 0 => x = 5
- x + 1 = 0 => x = -1
So, our critical points are x = 5 and x = -1. Think of these points as the boundaries of our solution intervals. They're like the checkpoints on our journey to solving the inequality.
Step 3: Create a Sign Chart
Now comes the fun part – creating a sign chart! This chart helps us visualize where the expression (x - 5)(x + 1) is positive, negative, or zero. Draw a number line and mark our critical points, -1 and 5. These points divide the number line into three intervals: (-∞, -1), (-1, 5), and (5, ∞).
Next, we'll pick a test value from each interval and plug it into our factored expression. This will tell us the sign of the expression in that interval:
- Interval (-∞, -1): Let's pick x = -2
- (-2 - 5)(-2 + 1) = (-7)(-1) = 7 (positive)
- Interval (-1, 5): Let's pick x = 0
- (0 - 5)(0 + 1) = (-5)(1) = -5 (negative)
- Interval (5, ∞): Let's pick x = 6
- (6 - 5)(6 + 1) = (1)(7) = 7 (positive)
Sign charts might seem a bit abstract at first, but they're lifesavers. They give us a bird's-eye view of how the expression behaves across different intervals. It's like having a map to guide us through the solution landscape.
Step 4: Determine the Solution Set
We're looking for where (x - 5)(x + 1) ≤ 0. This means we want the intervals where the expression is negative or zero. Looking at our sign chart, the expression is negative in the interval (-1, 5). It's also equal to zero at our critical points, x = -1 and x = 5. Since our inequality includes “equal to” (≤), we include these points in our solution.
Step 5: Express the Solution in Interval Notation
Finally, we can express our solution in interval notation. We include the endpoints -1 and 5 because the inequality is less than or equal to zero. So, our solution is:
[-1, 5]
Interval notation is like a mathematical shorthand. It's a concise way to communicate a whole range of solutions. The square brackets tell us that the endpoints are included, which is important in this case!
Wrapping Up
And there you have it! We've successfully solved the inequality x^2 - 4x - 5 ≤ 0 and expressed the solution in interval notation. Remember, the key is to factor the quadratic, find the critical points, create a sign chart, and then use the chart to determine the intervals that satisfy the inequality. It might seem like a lot of steps, but with a little practice, it becomes second nature.
So, next time you're faced with a quadratic inequality, don't sweat it! Just follow these steps, and you'll be solving them like a pro in no time. Keep practicing, and you'll be amazed at what you can achieve!