Solving The Inequality: Y < (2/3)x - 6
Hey guys! Let's dive into solving inequalities, specifically the one given: y < (2/3)x - 6. This type of problem is super common in algebra, and mastering it will definitely boost your math skills. We’ll break down the steps and then check each of the provided options to see which one fits the solution.
Understanding Linear Inequalities
Before we jump into the specific problem, let's make sure we're all on the same page about linear inequalities. Think of a linear inequality like a regular linear equation (y = mx + b), but instead of an equals sign, we have an inequality sign (<, >, ≤, or ≥). This means we’re not just looking for a single solution, but rather a range of solutions. The inequality y < (2/3)x - 6 represents all the points (x, y) on the coordinate plane that fall below the line y = (2/3)x - 6. This is because the “less than” symbol indicates that we are interested in the region where the y-values are smaller than the values calculated by the expression (2/3)x - 6. When visualizing this on a graph, you'd draw a dashed line at y = (2/3)x - 6 (dashed because the inequality is strict, meaning the points on the line are not included) and shade the area below the line. This shaded area represents all the possible solutions to the inequality. Understanding this graphical representation can make solving these problems much easier. Each point in the shaded region, when its x and y coordinates are plugged into the inequality, will make the statement true. Conversely, any point outside the shaded region will not satisfy the inequality. This concept is fundamental not just for solving simple inequalities but also for more complex systems of inequalities and linear programming problems. So, grasping this visual aspect early on can be a significant advantage in your mathematical journey. Remember, the dashed line signifies exclusion, while a solid line would indicate that the points on the line are also included in the solution set (for inequalities with ≤ or ≥).
Step-by-Step Solution
To figure out which point is a solution, we'll simply plug the x and y coordinates of each option into the inequality and see if it holds true. This is a straightforward method, and it’s really effective for this type of problem. Let's go through each option one by one:
A. (2, 11/16)
Plug in x = 2 and y = 11/16 into the inequality:
11/16 < (2/3) * 2 - 6
11/16 < 4/3 - 6
To compare these, we need a common denominator. Let's use 48:
(11 * 3) / 48 < (4 * 16) / 48 - (6 * 48) / 48
33/48 < 64/48 - 288/48
33/48 < -224/48
This statement is false, so (2, 11/16) is NOT a solution.
B. (0, -6)
Plug in x = 0 and y = -6:
-6 < (2/3) * 0 - 6
-6 < 0 - 6
-6 < -6
This statement is false because -6 is not less than -6; it is equal to -6. Remember, the inequality is strict (y < ...), so the points on the line are not solutions.
C. (3, 4)
Plug in x = 3 and y = 4:
4 < (2/3) * 3 - 6
4 < 2 - 6
4 < -4
This statement is false. So, (3, 4) is NOT a solution.
D. (1, -18/3)
Simplify -18/3 to -6, so we have the point (1, -6).
Plug in x = 1 and y = -6:
-6 < (2/3) * 1 - 6
-6 < 2/3 - 6
To compare, let's convert 6 to a fraction with a denominator of 3:
-6 < 2/3 - 18/3
-6 < -16/3
Now, convert -6 to a fraction with a denominator of 3:
-18/3 < -16/3
This statement is true! -18/3 is indeed less than -16/3. Therefore, (1, -6) is a solution.
Why Plugging in Points Works
This method of plugging in points is based on the fundamental definition of a solution to an inequality. A solution to an inequality is any ordered pair (x, y) that makes the inequality a true statement. When we substitute the x and y values of a point into the inequality, we are essentially checking whether that point satisfies the condition set by the inequality. If the resulting statement is true, the point is a solution; if it's false, the point is not a solution. This approach is incredibly versatile and can be used for any inequality, regardless of its complexity. For linear inequalities, like the one we're working with, the solutions form a region on the coordinate plane. This region is bounded by a line (or a dashed line for strict inequalities). The points within this region, and sometimes on the boundary line itself (for inequalities with ≤ or ≥), are the solutions. By plugging in points, we are effectively testing whether those points fall within the solution region. This method is also crucial for understanding the concept of solution sets and how they relate to the graphical representation of inequalities. It provides a concrete way to connect the algebraic expression of an inequality to its visual representation on the coordinate plane, making it a powerful tool for solving problems and building a deeper understanding of mathematical concepts. Furthermore, this approach is not limited to two-variable inequalities; it can be extended to inequalities with more variables, although the visualization becomes more challenging in higher dimensions.
Final Answer
So, after checking all the options, we found that the correct solution is D. (1, -18/3), which simplifies to (1, -6). This means that the point (1, -6) lies in the solution region of the inequality y < (2/3)x - 6.
Tips for Solving Inequalities
Before we wrap up, let’s cover some essential tips that can help you ace inequality problems every time. Remember, inequalities are a core concept in math, and mastering them will not only help you in exams but also in real-world problem-solving. The key is to approach them methodically and with a clear understanding of the underlying principles.
- Understand the Inequality Symbols: First off, make sure you're super clear on what each inequality symbol means. '<' means 'less than,' '>' means 'greater than,' '≤' means 'less than or equal to,' and '≥' means 'greater than or equal to.' Knowing these symbols like the back of your hand is crucial because they determine the range of solutions and how you represent them on a number line or a coordinate plane. For instance, strict inequalities (like < and >) will have dashed lines when graphing, indicating that the points on the line are not included in the solution. Non-strict inequalities (≤ and ≥), on the other hand, will have solid lines, showing that the boundary line is part of the solution. This distinction is vital for accurately interpreting and expressing the solution set.
- Treat Inequalities Like Equations (Mostly): For the most part, you can manipulate inequalities just like you would equations. You can add, subtract, multiply, or divide both sides to isolate the variable. However, there's one crucial exception: When you multiply or divide both sides by a negative number, you must flip the inequality sign. This is a critical rule to remember, as forgetting to flip the sign will lead to an incorrect solution. The reason for this flip is rooted in the number line's properties; multiplying or dividing by a negative number reverses the order of the numbers. For example, if 2 < 4, multiplying both sides by -1 gives -2 and -4. To maintain the truth of the statement, we need to flip the inequality sign to -2 > -4. This principle applies universally across all inequalities, so keep it at the forefront of your mind.
- Graphing Inequalities: Graphing inequalities can make visualizing the solution set much easier. For linear inequalities in two variables (like the one we solved), start by graphing the boundary line as if it were an equation (replace the inequality sign with an equals sign). Then, decide whether the line should be solid (for ≤ and ≥) or dashed (for < and >). Next, you need to determine which side of the line to shade. This shaded region represents all the solutions to the inequality. To figure out which side to shade, you can use a test point. Pick any point that is not on the line (for example, (0,0) is often a good choice). Plug the coordinates of the test point into the original inequality. If the inequality is true, shade the side of the line that contains the test point. If the inequality is false, shade the opposite side. This method provides a straightforward way to identify the solution region and visualize the infinite number of solutions.
- Checking Your Solution: Always, always, always check your solution! This is the golden rule of problem-solving in mathematics. For inequalities, this often involves plugging in a test point from your solution set back into the original inequality. If the inequality holds true for your test point, you can be confident that your solution is correct. If it doesn't, double-check your steps to find any errors. Checking your solution not only confirms the correctness of your answer but also reinforces your understanding of the inequality and its solution set. It's a simple yet powerful habit that can save you from making careless mistakes and boost your overall confidence in solving inequality problems.
By keeping these tips in mind and practicing regularly, you'll be solving inequalities like a pro in no time! Remember, math is all about understanding the concepts and applying them consistently. You've got this!
Practice Makes Perfect
The best way to get comfortable with inequalities is to practice! Try solving similar problems and varying the complexity. You can even create your own inequalities and solve them. The more you practice, the more confident you'll become. So, keep practicing, and don't be afraid to ask for help when you need it.