Solving Inequalities: Interval Method Examples

Hey guys! Today, we're diving deep into the interval method, a super useful technique for solving inequalities. Inequalities might seem tricky at first, but with the interval method, they become much more manageable. We'll walk through a couple of examples step-by-step, so you’ll be solving these like a pro in no time! So, let's get started and break down this powerful method together. We’ll tackle two specific inequalities: (x+4)(x-9) > 0 and (x+3)(4-x)(x+10) < 0. Let's jump right in!

Understanding the Interval Method

The interval method is a systematic way to solve inequalities, especially those involving polynomials or rational functions. The basic idea is to find the critical points where the expression changes sign and then test intervals between these points to determine where the inequality holds true. By using a number line and testing points, we can visually and logically determine the solution sets. It's a straightforward yet powerful technique. Think of it as creating a map of where the expression is positive, negative, or zero.

Key Steps in the Interval Method

Before we dive into the examples, let's outline the key steps involved in the interval method. Knowing these steps will make the process much clearer and easier to follow. Trust me, once you get the hang of these steps, inequalities will become a piece of cake! Here’s a quick rundown:

1. Find the Critical Points: Identify the values of x that make the expression equal to zero or undefined. These are your critical points. Setting each factor to zero and solving for x typically gives critical points.
2. Create Intervals: Use the critical points to divide the number line into intervals. Each interval represents a range of x-values that we'll test.
3. Test Each Interval: Pick a test value within each interval and plug it into the original inequality. Determine whether the inequality holds true or false for that interval. The choice of test value is arbitrary, any number within the interval will do.
4. Determine the Solution Set: Based on the test results, identify the intervals that satisfy the inequality. These intervals make up the solution set. Make sure to consider whether the endpoints of the intervals should be included (if the inequality includes equality) or excluded (if the inequality is strict).

Now that we’ve got the basic steps down, let’s put them into action with our first example. Get ready to see how this works in practice!

Example 1: Solving (x+4)(x-9) > 0

Alright, let's kick things off with our first example: (x+4)(x-9) > 0. We're going to walk through each step of the interval method to solve this inequality. Ready? Let’s go!

Step 1: Find the Critical Points

Our first mission is to find the critical points. Remember, these are the values of x that make the expression equal to zero. So, we need to set each factor to zero and solve for x:

• x + 4 = 0 => x = -4
• x - 9 = 0 => x = 9

So, our critical points are x = -4 and x = 9. These are the key turning points on our number line. They're like the boundaries that will help us define our intervals. These points are crucial because they are where the expression (x+4)(x-9) can change its sign (from positive to negative or vice versa).

Step 2: Create Intervals

Now that we have our critical points, we'll use them to divide the number line into intervals. Our critical points are -4 and 9, so we have three intervals:

• (-∞, -4)
• (-4, 9)
• (9, ∞)

Each of these intervals represents a range of x-values. We're going to test each interval to see if it satisfies our inequality. Think of these intervals as different zones on the number line, and we're figuring out which zones make our inequality true.

Step 3: Test Each Interval

This is where the fun begins! We need to pick a test value within each interval and plug it into our original inequality (x+4)(x-9) > 0. Let’s break it down:

• Interval (-∞, -4): Let's pick x = -5.
  • (-5 + 4)(-5 - 9) = (-1)(-14) = 14 > 0. So, the inequality holds true in this interval.
• Interval (-4, 9): Let's pick x = 0.
  • (0 + 4)(0 - 9) = (4)(-9) = -36 < 0. So, the inequality does not hold true in this interval.
• Interval (9, ∞): Let's pick x = 10.
  • (10 + 4)(10 - 9) = (14)(1) = 14 > 0. So, the inequality holds true in this interval.

We've essentially tested a representative from each interval to see if it makes the inequality true. This gives us a clear picture of which parts of the number line work for our solution.

Step 4: Determine the Solution Set

Based on our test results, we know that the inequality (x+4)(x-9) > 0 holds true for the intervals (-∞, -4) and (9, ∞). Since the inequality is strict (i.e., > and not ≥), we do not include the critical points in our solution set. Therefore, our solution set is:

• (-∞, -4) ∪ (9, ∞)

This means that any x-value less than -4 or greater than 9 will satisfy the inequality. And there you have it! We've solved our first inequality using the interval method. Feel like a pro yet? Let’s move on to the next example to solidify our understanding.

Example 2: Solving (x+3)(4-x)(x+10) < 0

Okay, now let’s tackle our second example: (x+3)(4-x)(x+10) < 0. This one’s a bit more complex since we have three factors, but don't worry, we'll use the same interval method steps to break it down. Ready to see how it’s done?

Step 1: Find the Critical Points

First up, we need to find the critical points. Just like before, these are the x-values that make the expression equal to zero. We’ll set each factor to zero and solve for x:

• x + 3 = 0 => x = -3
• 4 - x = 0 => x = 4
• x + 10 = 0 => x = -10

So, our critical points are x = -10, x = -3, and x = 4. These points are like the milestones that divide our number line into different sections. They’re where the sign of the expression can potentially change.

Step 2: Create Intervals

With our critical points in hand, we can now divide the number line into intervals. We have three critical points, which will give us four intervals:

• (-∞, -10)
• (-10, -3)
• (-3, 4)
• (4, ∞)

Each interval represents a different range of x-values that we need to test. Think of it as breaking down a big problem into smaller, manageable chunks. We’re systematically going through each range to see if it fits our inequality.

Step 3: Test Each Interval

Now comes the testing phase! We'll pick a test value within each interval and plug it into our original inequality (x+3)(4-x)(x+10) < 0. Let's go through each one:

• Interval (-∞, -10): Let's pick x = -11.
  • (-11 + 3)(4 - (-11))(-11 + 10) = (-8)(15)(-1) = 120 > 0. The inequality does not hold true.
• Interval (-10, -3): Let's pick x = -4.
  • (-4 + 3)(4 - (-4))(-4 + 10) = (-1)(8)(6) = -48 < 0. The inequality holds true.
• Interval (-3, 4): Let's pick x = 0.
  • (0 + 3)(4 - 0)(0 + 10) = (3)(4)(10) = 120 > 0. The inequality does not hold true.
• Interval (4, ∞): Let's pick x = 5.
  • (5 + 3)(4 - 5)(5 + 10) = (8)(-1)(15) = -120 < 0. The inequality holds true.

By testing these values, we’re getting a clear picture of which intervals satisfy our inequality. It’s like checking each piece of the puzzle to see if it fits.

Step 4: Determine the Solution Set

Looking at our test results, the inequality (x+3)(4-x)(x+10) < 0 holds true for the intervals (-10, -3) and (4, ∞). Since our inequality is strict (less than), we don't include the critical points in our solution. So, our solution set is:

• (-10, -3) ∪ (4, ∞)

This means that any x-value between -10 and -3, or greater than 4, will satisfy the inequality. Boom! We’ve successfully solved another inequality using the interval method. You’re getting the hang of it, right?

Tips and Tricks for the Interval Method

Before we wrap up, let's go over a few tips and tricks that can make using the interval method even easier. These little nuggets of wisdom can save you time and help you avoid common mistakes. Trust me, they’re worth knowing!

• Simplify First: If possible, simplify the inequality before starting the interval method. This might involve factoring, combining like terms, or rearranging the expression. A simplified inequality is often easier to work with and reduces the chances of making errors.
• Double-Check Critical Points: Always double-check your critical points to make sure you haven’t missed any or made a mistake in your calculations. Critical points are the foundation of the interval method, so accuracy is key.
• Choose Easy Test Values: When testing intervals, pick test values that are easy to work with. Zero is often a good choice, but any number in the interval will work. The goal is to make the calculation as straightforward as possible.
• Consider Endpoint Inclusion: Pay close attention to whether the inequality includes equality (≤ or ≥) or is strict (< or >). If it includes equality, you’ll need to include the critical points in your solution set. If it’s strict, you’ll exclude them.
• Visualize with a Number Line: Drawing a number line and marking the intervals can be incredibly helpful, especially for more complex inequalities. It provides a visual representation of the problem and makes it easier to see the solution.

Conclusion

So there you have it, guys! We’ve walked through the interval method for solving inequalities, step by step, with examples to illustrate the process. Remember, the key is to find the critical points, create intervals, test those intervals, and then determine your solution set. With practice, this method will become second nature. Inequalities might have seemed daunting at first, but now you’ve got a powerful tool in your math arsenal!

Keep practicing, and you’ll be solving even the trickiest inequalities with confidence. Happy problem-solving!