Solving Inequalities: Find Ordered Pairs For Y > (4/5)x - 3

by SLV Team 60 views
Solving Inequalities: Find Ordered Pairs for y > (4/5)x - 3

Hey guys! Today, we're diving into the world of inequalities and ordered pairs. Specifically, we're going to tackle the question: Which ordered pairs satisfy the inequality y > (4/5)x - 3? This is a common type of problem in algebra, and understanding how to solve it is super important. We'll break it down step-by-step, so you'll be a pro in no time! Inequalities might seem tricky at first, but with a little practice, you'll be solving them like a math whiz. The key here is to understand what an inequality represents graphically and how to test ordered pairs to see if they fit the solution. So, let's get started and make sure we choose the two correct solutions!

Understanding Inequalities and Ordered Pairs

Before we jump into solving, let's make sure we're all on the same page with the basics. An inequality is a mathematical statement that compares two expressions using symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Unlike an equation, which has one specific solution, an inequality has a range of solutions. In our case, we have the inequality y > (4/5)x - 3, which means we're looking for all the points (x, y) where the y-value is greater than the expression (4/5)x - 3.

An ordered pair is simply a pair of numbers, (x, y), that represents a point on a coordinate plane. The first number, x, tells us how far to move horizontally from the origin (0, 0), and the second number, y, tells us how far to move vertically. When we're dealing with inequalities, ordered pairs can either be solutions to the inequality or not. A solution is an ordered pair that, when plugged into the inequality, makes the statement true. If the ordered pair does not satisfy the inequality, it is not a solution.

Think of the inequality y > (4/5)x - 3 as a rule. We need to find ordered pairs that follow this rule. For example, if we had the ordered pair (0, 0), we would plug in x = 0 and y = 0 into the inequality and see if it holds true. This is the fundamental concept we'll use to solve our problem. By testing different ordered pairs, we can identify which ones make the inequality true and which ones don't. This process might seem a bit tedious, but it's a solid way to understand the solution set of an inequality.

The Graphical Representation of y > (4/5)x - 3

Now, let's visualize what the inequality y > (4/5)x - 3 looks like on a graph. This will give us a better understanding of why certain ordered pairs are solutions and others aren't. The inequality represents a region on the coordinate plane, not just a single line. To graph the inequality, we first graph the related equation y = (4/5)x - 3. This is a linear equation, meaning it forms a straight line when graphed.

The equation y = (4/5)x - 3 is in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. In our case, the slope m is 4/5, and the y-intercept b is -3. The y-intercept tells us where the line crosses the y-axis, which is at the point (0, -3). The slope tells us the steepness and direction of the line. A slope of 4/5 means that for every 5 units we move to the right on the x-axis, we move 4 units up on the y-axis.

When we graph the line y = (4/5)x - 3, we need to decide whether to use a solid line or a dashed line. Since our inequality is y > (4/5)x - 3, which does not include the equals sign, we use a dashed line. A dashed line indicates that the points on the line itself are not solutions to the inequality. If the inequality were y ≥ (4/5)x - 3, we would use a solid line to indicate that the points on the line are included in the solution set.

After drawing the dashed line, we need to shade the region of the graph that represents the solutions to the inequality. Since we want y to be greater than (4/5)x - 3, we shade the region above the line. This shaded region represents all the ordered pairs (x, y) that satisfy the inequality. Any point in this shaded region, when plugged into the inequality, will make the statement true. Conversely, any point below the dashed line will make the inequality false. This graphical understanding is crucial for visually checking if an ordered pair is a solution.

How to Test Ordered Pairs

The most straightforward way to determine if an ordered pair is a solution to the inequality y > (4/5)x - 3 is to simply plug the x and y values into the inequality and see if the statement holds true. Let's walk through the process step-by-step.

  1. Identify the Ordered Pair: Suppose we have an ordered pair, for example, (5, 2). The first number, 5, is the x-value, and the second number, 2, is the y-value.
  2. Substitute the Values: Replace x and y in the inequality with the values from the ordered pair. So, y > (4/5)x - 3 becomes 2 > (4/5)(5) - 3.
  3. Simplify the Expression: Perform the arithmetic on the right side of the inequality. First, multiply (4/5) by 5, which gives us 4. Then, subtract 3 from 4, which gives us 1. So, the inequality becomes 2 > 1.
  4. Check if the Statement is True: Determine if the resulting inequality is true or false. In our example, 2 > 1 is a true statement.
  5. Conclusion: If the statement is true, the ordered pair is a solution to the inequality. If the statement is false, the ordered pair is not a solution. Since 2 > 1 is true, the ordered pair (5, 2) is a solution to the inequality y > (4/5)x - 3.

Let's try another example with the ordered pair (0, -4). Substituting x = 0 and y = -4 into the inequality, we get -4 > (4/5)(0) - 3. Simplifying the right side, we have -4 > -3. This statement is false, so the ordered pair (0, -4) is not a solution to the inequality.

By testing ordered pairs in this way, we can systematically identify the solutions to the inequality. This method is especially useful when we have multiple ordered pairs to check, as in the original question where we need to select the two correct answers.

Selecting the Two Correct Answers

Now that we understand how to test ordered pairs, let's apply this knowledge to the original question. We need to identify two ordered pairs that satisfy the inequality y > (4/5)x - 3. Typically, you would be given a list of ordered pairs to choose from. For the purpose of this explanation, let's consider a few hypothetical ordered pairs and determine whether they are solutions:

  • Ordered Pair A: (5, 2) – We already tested this one and found that it is a solution.
  • Ordered Pair B: (0, -4) – We also tested this one and found that it is not a solution.
  • Ordered Pair C: (10, 5) – Let's test this one: 5 > (4/5)(10) - 3. Simplifying, we get 5 > 8 - 3, which is 5 > 5. This statement is false because 5 is not greater than 5. So, (10, 5) is not a solution.
  • Ordered Pair D: (-5, -8) – Let's test this one: -8 > (4/5)(-5) - 3. Simplifying, we get -8 > -4 - 3, which is -8 > -7. This statement is false, so (-5, -8) is not a solution.
  • Ordered Pair E: (0, -2) – Let's test this one: -2 > (4/5)(0) - 3. Simplifying, we get -2 > -3. This statement is true, so (0, -2) is a solution.

Based on our testing, Ordered Pair A (5, 2) and Ordered Pair E (0, -2) are solutions to the inequality y > (4/5)x - 3. Therefore, these would be the two correct answers.

Remember, the key to solving these types of problems is to systematically test each ordered pair by plugging the x and y values into the inequality. If the resulting statement is true, the ordered pair is a solution. If it's false, the ordered pair is not a solution. With practice, you'll become quicker at recognizing which ordered pairs are likely to be solutions just by glancing at them in relation to the inequality.

Tips and Tricks for Solving Inequalities

To wrap things up, let's go over a few tips and tricks that can help you solve inequalities more efficiently:

  1. Visualize the Graph: Mentally picture the graph of the inequality. This can help you quickly eliminate ordered pairs that are clearly not solutions. Remember that y > (4/5)x - 3 represents the region above a dashed line, so ordered pairs far below the line are unlikely to be solutions.
  2. Look for Easy Points: Ordered pairs with zeros are often the easiest to test. For example, if an ordered pair has x = 0 or y = 0, the calculations will be simpler.
  3. Simplify Before Testing: If the inequality has fractions or decimals, it might be helpful to simplify it before testing ordered pairs. This can reduce the chance of making arithmetic errors.
  4. Use the Slope and Y-Intercept: Understanding the slope and y-intercept of the related equation y = (4/5)x - 3 can give you a sense of the direction and position of the line. This can help you predict which ordered pairs are more likely to be solutions.
  5. Double-Check Your Work: Always double-check your calculations to ensure you haven't made any mistakes. A small arithmetic error can lead to the wrong answer.

By following these tips and tricks, you'll be well-equipped to tackle inequality problems with confidence. Remember, practice makes perfect! The more you work with inequalities and ordered pairs, the easier it will become to identify the solutions.

Conclusion

So, there you have it! We've explored how to determine which ordered pairs are solutions to the inequality y > (4/5)x - 3. We covered the basics of inequalities, the graphical representation of the inequality, how to test ordered pairs, and some helpful tips and tricks. Remember, the key is to plug in the x and y values of the ordered pair into the inequality and see if the statement is true.

Understanding inequalities is a crucial skill in algebra and beyond. Whether you're solving problems in a math class or applying mathematical concepts in real-world situations, the ability to work with inequalities will serve you well. Keep practicing, and don't hesitate to review these steps whenever you encounter a similar problem. You've got this! Now go out there and conquer those inequalities!