Solving (x-5)(x+4) ≤ 0: A Step-by-Step Guide

Nov 19, 2025 by Andrew McMorgan 45 views

Solving Polynomial Inequalities: A Comprehensive Guide to (x-5)(x+4) ≤ 0

Hey math enthusiasts! Ever find yourself staring blankly at a polynomial inequality, wondering how to crack the code? Well, you're in the right place. Today, we're diving deep into solving the polynomial inequality (x-5)(x+4) ≤ 0. We’ll break down each step, from finding critical values to graphing the solution set and expressing it in interval notation. Trust me, by the end of this guide, you'll be a pro at tackling these types of problems. So, grab your pencils and let's get started!

Understanding Polynomial Inequalities

Before we jump into the nitty-gritty, let’s get a handle on what polynomial inequalities are all about. Polynomial inequalities are mathematical statements that compare a polynomial expression to a value (in our case, 0) using inequality symbols like ≤, ≥, <, or >. Solving these inequalities means finding the range of values for the variable (x, in our case) that make the inequality true. It's like finding the sweet spot where everything clicks into place. Think of it as a puzzle where the pieces are numbers, and you need to find the ones that fit just right.

Polynomial inequalities are super useful in various fields, from engineering to economics. They help us model real-world scenarios where quantities have upper and lower bounds. For instance, engineers might use them to design structures that can withstand certain loads, while economists might use them to predict price ranges. So, understanding how to solve these inequalities isn't just an academic exercise; it's a practical skill that can open doors in many areas.

Moreover, polynomial inequalities build upon the foundational concepts of algebra and calculus. They require a solid understanding of factoring, number lines, and interval notation – all of which are crucial for more advanced mathematical topics. So, mastering these inequalities now sets you up for success in future math courses and beyond. It's like building a strong foundation for a skyscraper; the higher you want to go, the sturdier the base needs to be.

Key Concepts in Solving Polynomial Inequalities

To really nail this, there are a few key concepts we need to keep in mind. First up are critical values. These are the values of x that make the polynomial expression equal to zero. They're like the turning points on a rollercoaster – the spots where things change direction. In our case, they're the values that make (x-5)(x+4) equal to 0. Finding these values is the first major step in solving the inequality.

Next, we need to understand how to use a number line. A number line is our visual tool for mapping out the solution set. We mark the critical values on the number line, which divides it into intervals. Then, we test a value from each interval in the original inequality to see if it holds true. It's like a treasure map where the critical values are landmarks, and we're searching for the hidden solution within the intervals.

Lastly, we need to be fluent in interval notation. This is a concise way to express the solution set, using brackets and parentheses to indicate whether the endpoints are included or excluded. It's like a secret code that mathematicians use to communicate sets of numbers. For example, [a, b] means all numbers between a and b, including a and b, while (a, b) means all numbers between a and b, but not including a and b. Mastering interval notation is crucial for accurately expressing our solutions.

Step-by-Step Solution for (x-5)(x+4) ≤ 0

Alright, let's get down to business and solve the inequality (x-5)(x+4) ≤ 0 step by step. This is where the rubber meets the road, and we put our knowledge into action. Don't worry, we'll take it slow and explain each step in detail. By following along, you'll see how each concept fits together to form the complete solution.

Step 1: Find the Critical Values

The first thing we need to do is find the critical values. Remember, these are the values of x that make the polynomial expression equal to zero. In our case, that means we need to solve the equation (x-5)(x+4) = 0. This is where our factoring skills come into play. We have a product of two factors, and the product is zero if and only if at least one of the factors is zero. So, we set each factor equal to zero and solve for x:

x - 5 = 0 => x = 5
x + 4 = 0 => x = -4

So, our critical values are x = 5 and x = -4. These are the key points that will divide our number line into intervals. Think of them as the anchors that hold our solution in place. They're the turning points where the inequality might switch from being true to false, or vice versa. Without these critical values, we'd be wandering in the dark, unable to find our solution.

Step 2: Create a Number Line and Divide it into Intervals

Now that we have our critical values, it’s time to create a number line. Draw a straight line and mark our critical values, -4 and 5, on it. These points divide the number line into three intervals: (-∞, -4), (-4, 5), and (5, ∞). These intervals are like the different territories on our map, and we need to explore each one to see if it contains a solution to our inequality.

The number line is our visual aid, helping us to see how the critical values split the real numbers into different regions. Each interval represents a set of numbers that are either all solutions or all non-solutions to our inequality. It’s like a buffet table where each section offers a different dish – some tasty, some not so much. Our job is to figure out which sections are worth sampling.

Step 3: Test a Value from Each Interval in the Original Inequality

This is where the real investigation begins. We need to test a value from each interval in the original inequality, (x-5)(x+4) ≤ 0, to see if the inequality holds true. This will tell us which intervals contain solutions. Think of it as a taste test – we’re sampling a number from each interval to see if it satisfies our criteria.

Let's pick a test value from each interval:

For the interval (-∞, -4), let's pick x = -5.
For the interval (-4, 5), let's pick x = 0.
For the interval (5, ∞), let's pick x = 6.

Now, we plug each of these values into the inequality and see what happens:

For x = -5: (-5-5)(-5+4) = (-10)(-1) = 10. Is 10 ≤ 0? No.
For x = 0: (0-5)(0+4) = (-5)(4) = -20. Is -20 ≤ 0? Yes!
For x = 6: (6-5)(6+4) = (1)(10) = 10. Is 10 ≤ 0? No.

Step 4: Determine the Solution Set

Based on our testing, we've found that the inequality (x-5)(x+4) ≤ 0 holds true for the interval (-4, 5). But remember, our inequality includes “≤”, which means we also need to consider the critical values themselves. Since the inequality is less than or equal to zero, the critical values, where the expression equals zero, are also part of the solution. It's like adding the final touches to our masterpiece – the critical values complete the picture.

So, the solution set includes the interval (-4, 5) and the points x = -4 and x = 5. This is a crucial point – if the inequality had been (x-5)(x+4) < 0, we would exclude the critical values. But because we have the “equal to” part, we embrace them as part of our solution. It’s a subtle but significant detail that can make all the difference.

Step 5: Express the Solution Set in Interval Notation and Graph it

Finally, we’re ready to express our solution set in interval notation and graph it on a real number line. This is the grand finale, where we present our findings in a clear and concise way. Think of it as writing the conclusion to our mathematical detective story.

In interval notation, we use brackets [ ] to include the endpoints and parentheses ( ) to exclude them. Since we're including both -4 and 5, we use brackets. So, the solution set in interval notation is [-4, 5]. This notation tells us that the solution includes all numbers between -4 and 5, as well as -4 and 5 themselves. It's a neat and efficient way to communicate our solution.

To graph the solution set, we draw a number line and mark -4 and 5 with closed circles (or brackets) to indicate that they are included in the solution. Then, we shade the region between -4 and 5 to represent all the numbers in the interval. The graph provides a visual representation of our solution, making it easy to see the range of values that satisfy the inequality. It’s like a map that shows us exactly where our treasure lies.

Graphing the Solution Set

To visualize our solution, let's graph the solution set on a real number line. This is like creating a visual aid that helps us see the range of values that satisfy the inequality. A graph can often make a complex solution much easier to understand at a glance. Think of it as turning numbers into a picture – sometimes, seeing is believing.

Steps to Graph the Solution Set

Draw a number line: Start by drawing a straight line and marking the critical values, -4 and 5, on it. These are our reference points, the anchors that hold our solution in place.
Use brackets or parentheses: Since our inequality includes “≤”, we use closed brackets [ ] at -4 and 5 to indicate that these values are included in the solution set. If we had a strict inequality (< or >), we would use open parentheses ( ) to show that the endpoints are not included. This is a crucial visual cue that tells us whether the endpoints are part of the solution.
Shade the interval: Shade the region between -4 and 5 to represent all the numbers in the interval [-4, 5]. This shaded region is the heart of our solution, the range of values that make the inequality true. It’s like highlighting the path to the treasure on our map.

By graphing the solution set, we create a clear and intuitive representation of our answer. The number line becomes a visual story, showing us exactly which values satisfy the inequality. It’s a powerful tool for understanding and communicating mathematical solutions.

Expressing the Solution Set in Interval Notation

Now, let's express our solution set in interval notation. This is a concise and standard way to represent a set of numbers, using symbols to indicate the range and whether the endpoints are included or excluded. It’s like learning a new language – once you’re fluent in interval notation, you can communicate mathematical solutions with precision and ease.

Understanding Interval Notation

Interval notation uses parentheses ( ) and brackets [ ] to denote intervals of numbers. Parentheses indicate that the endpoint is not included, while brackets indicate that the endpoint is included. For example:

(a, b) represents all numbers between a and b, but not including a and b.
[a, b] represents all numbers between a and b, including a and b.
(a, ∞) represents all numbers greater than a.
(-∞, b) represents all numbers less than b.

In our case, the solution set includes all numbers between -4 and 5, as well as -4 and 5 themselves. So, we use brackets to indicate that the endpoints are included. The interval notation for our solution set is [-4, 5]. This is a compact and unambiguous way to convey our solution, making it easy for others to understand the range of values that satisfy the inequality.

Conclusion

So, there you have it! We’ve successfully solved the polynomial inequality (x-5)(x+4) ≤ 0, graphed the solution set on a real number line, and expressed it in interval notation. You've navigated the twists and turns of polynomial inequalities like a pro. Remember, the key is to break down the problem into manageable steps, find the critical values, test intervals, and then express the solution clearly.

This journey through solving (x-5)(x+4) ≤ 0 is more than just an academic exercise. It's a testament to your problem-solving skills and your ability to tackle mathematical challenges head-on. The concepts we've covered today – critical values, number lines, interval notation – are not just tools for solving inequalities; they're foundational concepts that will serve you well in more advanced math courses and beyond. So, keep practicing, keep exploring, and never stop challenging yourself.

Keep practicing, and you'll be mastering polynomial inequalities in no time. You guys got this!