Solving $x^2 - 2x - 8 > 0$ Using The F(x) > 0 Method Cut Points And Intervals

Hey guys! Today, we're diving into the exciting world of solving inequalities, specifically focusing on the type $f(x) > 0$. We're going to break down the inequality $x^2 - 2x - 8 > 0$, using a super cool interactive figure to help us visualize what's going on. So buckle up, and let's get started!

Part 1 Unveiling the Cut Points

Discovering the Critical Values

Okay, so first things first, we need to check the "Show Cut Points" option. These cut points, also known as critical values or roots, are the key to unlocking the solution. These points are where the function $f(x)$ actually equals zero, which means they're the boundaries where the function might switch from being positive to negative, or vice versa. Think of them as the dividing lines on a number line that separate the regions where our inequality holds true or false. To find these cut points, we essentially need to solve the equation $x^2 - 2x - 8 = 0$.

Now, there are a couple of ways we can tackle this quadratic equation. We could use the classic quadratic formula, which is a reliable method for any quadratic equation. But in this case, we can actually make our lives a little easier by factoring. Factoring is like reverse-distributing, and it's a great way to simplify things when it's possible. So, let's see if we can factor $x^2 - 2x - 8$. We're looking for two numbers that multiply to -8 and add up to -2. After a little bit of thought, we can see that -4 and 2 fit the bill perfectly. Therefore, we can factor the quadratic as $(x - 4)(x + 2) = 0$.

Great! Now that we've factored the equation, finding the cut points is a breeze. For the product of two factors to be zero, at least one of them has to be zero. So, we set each factor equal to zero and solve for $x$. This gives us $x - 4 = 0$ or $x + 2 = 0$. Solving these simple equations, we find that $x = 4$ and $x = -2$. These are our cut points! These are super important values because they divide the number line into intervals where the function’s sign remains constant. The cut points are the anchors that define the solution sets for the inequality. These numbers essentially act as the boundaries where the graph of our quadratic function crosses the x-axis. By identifying them, we can easily determine the intervals where the function is positive or negative.

Visualizing the Number Line

So, what do these cut points actually mean in the context of our inequality? Well, imagine a number line stretching out infinitely in both directions. Our cut points, -2 and 4, mark specific locations on this line. These points divide the number line into three distinct intervals: $(-\infty, -2)$, $(-2, 4)$, and $(4, \infty)$. The beauty of these intervals is that the sign of our function $f(x) = x^2 - 2x - 8$ will remain consistent within each interval. It will either be positive throughout the entire interval, negative throughout, or neither (if the interval happens to include a cut point, where the function is zero). This is a fundamental concept in solving inequalities – the sign of a continuous function can only change at its roots (cut points).

By pinpointing these cut points, we've essentially laid the foundation for determining where our function is greater than zero. They are the crucial markers that help us break down the problem into manageable sections. We've identified the critical values that dictate the behavior of our inequality, making it much easier to find the solutions. Remember, the name of the game here is to find the values of $x$ that make $x^2 - 2x - 8$ strictly positive, and the cut points are our guides.

Part 2 Solving the Inequality

Testing the Intervals

Now that we've found our cut points and divided the number line into intervals, the next step is to determine whether the function $f(x) = x^2 - 2x - 8$ is positive or negative in each of these intervals. This is where the magic happens! We're going to use a technique called test points. This method is super straightforward – we simply pick a test value within each interval and plug it into our function. The sign of the result will tell us the sign of the function throughout the entire interval. Remember, the sign of the function can only change at the cut points, so whatever sign we find for our test point will hold true for the whole interval.

Let's start with the interval $(-\infty, -2)$. A convenient test point here would be $x = -3$. Plugging this into our function, we get $f(-3) = (-3)^2 - 2(-3) - 8 = 9 + 6 - 8 = 7$. Since 7 is positive, we know that $f(x)$ is positive throughout the entire interval $(-\infty, -2)$. This means that all the values of $x$ in this interval are solutions to our inequality $x^2 - 2x - 8 > 0$!

Next, let's move on to the interval $(-2, 4)$. A nice and easy test point to use here is $x = 0$. Plugging this in, we get $f(0) = (0)^2 - 2(0) - 8 = -8$. This is negative, so $f(x)$ is negative throughout the interval $(-2, 4)$. This means that none of the values of $x$ in this interval are solutions to our inequality.

Finally, let's consider the interval $(4, \infty)$. A good test point here would be $x = 5$. Plugging this in, we get $f(5) = (5)^2 - 2(5) - 8 = 25 - 10 - 8 = 7$. This is positive again, so $f(x)$ is positive throughout the interval $(4, \infty)$. This means that all the values of $x$ in this interval are also solutions to our inequality.

Writing the Solution

We've done the hard work of identifying the intervals where $f(x) > 0$. Now, all that's left is to express our solution in a clear and concise way. We know that the inequality holds true for the intervals $(-\infty, -2)$ and $(4, \infty)$. Since we're looking for values where $f(x)$ is strictly greater than zero, we don't include the cut points themselves in our solution (because at the cut points, $f(x) = 0$).

Therefore, the solution to the inequality $x^2 - 2x - 8 > 0$ is the union of these two intervals. We can write this in interval notation as $(-\infty, -2) \cup (4, \infty)$. This notation simply means "all the numbers from negative infinity up to (but not including) -2, combined with all the numbers from 4 (but not including) to positive infinity." And that's it! We've successfully solved the inequality using the power of cut points and test intervals.

The Big Picture

Let's recap what we've done. We started with an inequality, $x^2 - 2x - 8 > 0$, and our goal was to find all the values of $x$ that satisfy this inequality. We used a graphical approach, focusing on identifying the intervals where the function $f(x) = x^2 - 2x - 8$ is positive.

The first crucial step was to find the cut points, which are the x-values where the function equals zero. These points divide the number line into intervals. Then, we used test points within each interval to determine whether the function is positive or negative in that interval. By plugging these test points into our function, we found the intervals where $f(x) > 0$, which gave us our solution.

In the end, the solution to the inequality $x^2 - 2x - 8 > 0$ is $(-\infty, -2) \cup (4, \infty)$. This means that any value of $x$ less than -2 or greater than 4 will satisfy the inequality. The combination of cut points and test intervals is a powerful technique that can be applied to solve a wide range of inequalities. Remember, the key is to visualize the function and how its sign changes across the number line.

So, there you have it, guys! We've successfully navigated the world of inequalities and found the solution to $x^2 - 2x - 8 > 0$. This method of using cut points and test intervals is a fantastic tool to have in your mathematical toolkit. Keep practicing, and you'll be solving inequalities like a pro in no time!