Solving The Inequality: 5(x-2)(x+4) > 0

by Andrew McMorgan 40 views

Hey math enthusiasts! Today, we're diving into the world of inequalities and tackling a specific problem: finding the solution set for the inequality 5(x-2)(x+4) > 0. This might seem intimidating at first, but don't worry, we'll break it down step by step so you can conquer it with confidence. Let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the question is asking. We're given the inequality 5(x-2)(x+4) > 0, and our goal is to find all the values of x that make this statement true. In other words, we want to find the range of x values that, when plugged into the inequality, will result in a positive number. This range of values is called the solution set.

To find this solution set, we'll use a method that involves identifying critical points and testing intervals. These critical points are the values of x that make the expression on the left side of the inequality equal to zero. They are crucial because they divide the number line into intervals where the expression is either positive or negative. By testing a value from each interval, we can determine which intervals satisfy the inequality.

Think of it like this: imagine a rollercoaster. The critical points are like the peaks and valleys of the ride. Between these points, the rollercoaster is either going up or down. Similarly, between our critical points, the expression 5(x-2)(x+4) is either positive or negative. Our job is to find the sections of the ride (intervals) where the rollercoaster (expression) is above zero (positive).

This method is a powerful tool for solving a wide range of inequalities, so mastering it will definitely boost your math skills. Now, let's roll up our sleeves and get to the actual solving!

Step 1: Finding the Critical Points

Okay, guys, let's get down to business! The first step in solving this inequality is to find the critical points. Remember, these are the values of x that make the expression on the left side of the inequality equal to zero. So, we need to solve the equation:

5(x-2)(x+4) = 0

This equation is already factored, which makes our lives much easier! A product of factors is equal to zero if and only if at least one of the factors is equal to zero. This is a fundamental principle in algebra, and it's super helpful for solving equations like this.

So, we set each factor equal to zero and solve for x:

  • x - 2 = 0 => x = 2
  • x + 4 = 0 => x = -4

Great! We've found our critical points: x = 2 and x = -4. These points are like the landmarks on our number line, dividing it into different regions that we need to investigate.

Now, let's think about what these critical points represent. They are the values where the expression 5(x-2)(x+4) changes its sign. To the left of -4, the expression will have one sign, and to the right of -4, it might have a different sign. The same goes for the point 2. This is because at these points, one of the factors changes from negative to positive or vice versa.

Finding these critical points is a crucial step because it allows us to break down the problem into smaller, more manageable chunks. Instead of dealing with the entire number line, we can focus on the intervals created by these points. This makes the process of determining the solution set much more straightforward. So, with our critical points in hand, we're ready to move on to the next step: testing the intervals!

Step 2: Testing the Intervals

Alright, we've found our critical points: x = -4 and x = 2. Now, it's time to put them to work and see what they tell us about the inequality 5(x-2)(x+4) > 0. These critical points divide the number line into three distinct intervals:

  1. x < -4
  2. -4 < x < 2
  3. x > 2

Our next task is to determine whether the expression 5(x-2)(x+4) is positive or negative in each of these intervals. To do this, we'll choose a test value from each interval and plug it into the expression. The sign of the result will tell us whether the entire interval satisfies the inequality or not.

Let's start with the first interval, x < -4. A convenient test value would be x = -5. Plugging this into our expression, we get:

5((-5)-2)((-5)+4) = 5(-7)(-1) = 35

Since 35 is positive, the expression 5(x-2)(x+4) is positive for all values of x in the interval x < -4. This means this interval is part of our solution set!

Now, let's move on to the second interval, -4 < x < 2. A good test value here would be x = 0. Plugging this in, we get:

5(0-2)(0+4) = 5(-2)(4) = -40

Since -40 is negative, the expression is negative in this interval. This means the interval -4 < x < 2 is not part of our solution set.

Finally, let's test the third interval, x > 2. We can use x = 3 as our test value:

5(3-2)(3+4) = 5(1)(7) = 35

Again, we get a positive result, so the expression is positive in the interval x > 2. This interval is part of our solution set!

By testing these intervals, we've effectively mapped out where the expression 5(x-2)(x+4) is positive and where it's negative. This is like having a treasure map that shows us exactly where to find the solutions to our inequality. Now, let's put all this information together and write down the solution set.

Step 3: Writing the Solution Set

Okay, team, we've done the hard work! We've found the critical points, tested the intervals, and now we're ready to write down the solution set for the inequality 5(x-2)(x+4) > 0. Remember, the solution set includes all the values of x that make the inequality true.

From our interval testing, we found that the expression 5(x-2)(x+4) is positive in two intervals:

  1. x < -4
  2. x > 2

So, the solution set consists of all x values that are either less than -4 or greater than 2. We can write this using interval notation or set-builder notation. Let's use both to make sure we're crystal clear.

In interval notation, we use parentheses and infinity symbols to represent the intervals. Parentheses indicate that the endpoint is not included in the solution set (because the inequality is strictly greater than 0, not greater than or equal to 0). So, the solution set in interval notation is:

(-∞, -4) ∪ (2, ∞)

The symbol ∪ means "union," which indicates that we're combining these two intervals into a single solution set.

In set-builder notation, we use a more formal notation to describe the set of solutions. We write:

{x | x < -4 or x > 2}

This is read as "the set of all x such that x is less than -4 or x is greater than 2." Both notations are perfectly valid ways to represent the solution set, so choose the one you're most comfortable with.

We've successfully found the solution set to the inequality 5(x-2)(x+4) > 0. Give yourselves a pat on the back! This is a great example of how to use critical points and interval testing to solve inequalities. Now, let's recap the key steps we took to conquer this problem.

Conclusion: Key Takeaways

Awesome job, guys! We've navigated the world of inequalities and successfully found the solution set for 5(x-2)(x+4) > 0. Let's quickly recap the key steps we took so you can apply this method to other problems:

  1. Find the Critical Points: Set the expression equal to zero and solve for x. These points are the boundaries of our intervals.
  2. Test the Intervals: Choose a test value from each interval created by the critical points and plug it into the original inequality. The sign of the result tells you whether the interval is part of the solution set.
  3. Write the Solution Set: Express the solution using either interval notation or set-builder notation.

Remember, practice makes perfect! The more you work with inequalities, the more comfortable you'll become with this process. So, don't be afraid to tackle new challenges and keep honing your skills. You've got this!

Understanding inequalities is a fundamental concept in mathematics, and mastering it will open doors to more advanced topics. Whether you're dealing with quadratic inequalities, rational inequalities, or even inequalities involving absolute values, the principles we've discussed today will serve you well.

So, keep exploring, keep learning, and most importantly, keep having fun with math! You've just added another tool to your mathematical toolkit, and that's something to be proud of. Until next time, happy problem-solving!