Solving Inequalities: A Step-by-Step Guide

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Solving Inequalities and Graphing Solutions: A Comprehensive Guide

Hey math enthusiasts! Today, we're diving into the world of inequalities, specifically how to solve them and visualize the solutions by graphing them. We'll break down the process step-by-step, making it super easy to understand. So, grab your pencils and let's get started!

Understanding Inequalities: The Basics

Alright guys, before we jump into solving, let's make sure we're all on the same page about what inequalities actually are. In math, an inequality is a statement that compares two values, showing that they are not equal. Instead of an equals sign (=), we use symbols like:

  • < : Less than
  • > : Greater than
  • ≤ : Less than or equal to
  • ≥ : Greater than or equal to

These symbols help us express relationships where one value is bigger or smaller than another, or possibly equal. Think of it like a seesaw. If both sides are equal, the seesaw is balanced. But if one side is heavier (greater), the seesaw tips in that direction. Inequalities help us describe that imbalance. Now, in the case of our example, we are looking at −5y<−20-5y < -20. This reads as "negative five times y is less than negative twenty." The goal is to find all the values of 'y' that make this statement true. This means, we're not just looking for a single answer, but rather a range of answers.

Working with inequalities is similar to working with regular equations, but there's a crucial twist that you need to remember. We'll get to that in a bit. The main idea is to isolate the variable (in our case, 'y') on one side of the inequality. We do this by performing operations on both sides, just like with equations. However, one rule is going to be incredibly important for you to remember. Now, let's explore how to solve an inequality step-by-step. Let's make sure we understand the logic before we begin the solving process. For instance, if you have a number line, imagine you're standing on the number '0'. Numbers to the right of you are greater than zero, while numbers to the left are less than zero. This concept will be very important when we begin to graph our answers, guys. We must remember that our goal is to isolate the variable. We're going to use the properties of inequalities to simplify and find the solution set. Keep in mind that the solution set is the set of all the values of the variable that make the inequality true. Ready? Let's dive in!

Solving the Inequality: −5y<−20-5y < -20

Alright, let's get down to the nitty-gritty and solve the inequality -5y < -20.

  1. Isolate the Variable: Our primary objective is to get 'y' by itself on one side of the inequality. To do this, we need to get rid of the -5 that's multiplying 'y'. We do this by dividing both sides of the inequality by -5. So, we'll perform the division: (-5y) / -5 < -20 / -5.
  2. The Golden Rule: Flipping the Sign: This is the important part I mentioned earlier! When you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. This is because multiplying or dividing by a negative number reverses the order on the number line. For instance, think about it like this: 2 is less than 3. But if we multiply both sides by -1, we get -2 and -3. Suddenly, -2 is greater than -3. So, in our case, since we're dividing by -5 (a negative number), our '<' sign will become '>'.
  3. Simplify: Now, let's simplify our inequality. Dividing -5y by -5 gives us just 'y'. Dividing -20 by -5 gives us 4. So, our inequality becomes: y > 4.

Therefore, the solution to the inequality -5y < -20 is y > 4. This means any value of 'y' that is greater than 4 will make the original inequality true. We're not looking for a single number; we're looking for an infinite set of numbers.

Now, let's explore some ways to check our answers. Checking your work is an essential part of the process, guys. You want to make sure your answer is correct. Let's pick a number that is greater than 4. How about the number 5? If we plug in 5 for the variable 'y' in the original problem, we'll get -5 times 5, which is -25. Is -25 less than -20? Absolutely! That's correct! Now, let's try a number that is less than 4, such as 3. Plug that in, and you will get -15. Is -15 less than -20? No, it's not. Therefore, that shows the correct range of numbers that we are looking for. Now that we understand how to solve the inequality and check the answer, let's get into the graphing portion.

Graphing the Solution Set

Alright, let's graph the solution set. Graphing helps us visually represent all the possible values that satisfy the inequality. Here's how:

  1. Draw a Number Line: Draw a number line. Make sure it extends far enough to include the critical value (in our case, 4) and some numbers on either side of it. Include a few numbers above and below your answer. For instance, include 2, 3, 4, 5, 6, and 7 on the number line.
  2. Mark the Critical Value: Locate the number 4 on the number line. Since our inequality is y > 4 (and not y ≥ 4), we use an open circle (or parenthesis) at 4. An open circle means that 4 is not included in the solution set. If the inequality was y ≥ 4, we would use a closed circle (or bracket) to indicate that 4 is included.
  3. Shade the Solution Area: Since 'y' is greater than 4, we need to shade the portion of the number line that includes all the numbers greater than 4. This means we shade to the right of the open circle at 4. The shaded region represents all the values that make the inequality true.

Your graph should show an open circle at 4, and an arrow going to the right, showing that the solution includes all numbers greater than 4. To give you some more context, imagine a car that can only go over 4 miles per hour. That means 4 mph is not an option; it must be greater. Therefore, the graph starts at the number 4, but it is not included. It continues on forever! Therefore, the solution is infinite.

Understanding the Graph

The graph is a visual representation of the solution set. It tells us at a glance all the values that satisfy the inequality. Everything to the right of the open circle at 4 is a solution. Any number you pick in that shaded area, when plugged back into the original inequality, will make the statement true. If the number falls outside the shaded area, it's not a solution.

  • Open vs. Closed Circles: Remember, an open circle means the critical value is not included in the solution, and a closed circle means it is included. The inequality sign tells us which to use. '>`' or '<' use an open circle. '≥' or '≤' use a closed circle.
  • Direction of Shading: The direction you shade indicates which side of the number line contains the solutions. If 'y > 4', you shade to the right (greater than). If 'y < 4', you shade to the left (less than).

Key Takeaways

Let's recap what we've learned, guys!

  • Inequalities: They compare values using symbols like <, >, ≤, and ≥.
  • Solving Inequalities: Isolate the variable using the same principles as solving equations. Flip the inequality sign when multiplying or dividing by a negative number.
  • Graphing Solutions: Draw a number line, mark the critical value with an open or closed circle (depending on the inequality sign), and shade the area representing the solutions.

Conclusion

Solving inequalities and graphing their solutions is a fundamental skill in mathematics. By following these steps, you can confidently solve and visualize a wide range of inequalities. Keep practicing, and you'll become a pro in no time! Remember to always pay attention to the inequality sign and whether you need to flip it. And don't forget the importance of checking your work! Keep up the great work! That's all for today, guys! Keep practicing, and you'll do great! And that's all, folks!