Solving Linear Inequalities Identifying Solutions For Y < 0.5x + 2
Hey guys! Today, we're diving into the world of linear inequalities. Specifically, we're going to tackle the question: Which points are solutions to the linear inequality y < 0.5x + 2? And to make things interesting, we've got five points to test out: (-3, -2), (-2, 1), (-1, -2), (-1, 2), and (1, -2). Buckle up, because we're about to solve this like pros!
Understanding Linear Inequalities
Before we jump into the points, let's quickly recap what linear inequalities are all about. A linear inequality is very similar to a linear equation, but instead of an equals sign (=), it uses inequality signs like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Think of it as a way to describe a range of possible solutions rather than just one specific solution. In our case, we're dealing with y < 0.5x + 2, which means we're looking for all the points (x, y) where the y-coordinate is strictly less than 0.5 times the x-coordinate, plus 2.
Graphically, a linear inequality represents a half-plane. The boundary line (in this case, y = 0.5x + 2) divides the coordinate plane into two regions. One of these regions contains the solutions to the inequality, and the other doesn't. The type of inequality sign tells us whether the boundary line itself is included in the solution set (≤ or ≥) or not (< or >). Since we have y < 0.5x + 2, the boundary line will be dashed (or dotted) to indicate that points on the line are not solutions. To find the solutions, we'll test each point to see if it falls into the correct region.
Testing the Points: The Key to Finding Solutions
Alright, let's get our hands dirty and test each point one by one. This is where the magic happens! We'll substitute the x and y coordinates of each point into the inequality y < 0.5x + 2 and see if the inequality holds true. If it does, that point is a solution. If not, it's not.
1. The Point (-3, -2)
First up, we have the point (-3, -2). This means x = -3 and y = -2. Let's plug these values into our inequality:
-2 < 0.5(-3) + 2 -2 < -1.5 + 2 -2 < 0.5
Is this true? Yes, -2 is indeed less than 0.5. So, the point (-3, -2) is a solution to the inequality!
2. The Point (-2, 1)
Next, we're testing the point (-2, 1), where x = -2 and y = 1. Let's substitute again:
1 < 0.5(-2) + 2 1 < -1 + 2 1 < 1
Hold on a second! Is 1 less than 1? Nope. 1 is equal to 1, but it's not less than 1. Therefore, the point (-2, 1) is not a solution.
3. The Point (-1, -2)
On to our third point: (-1, -2), with x = -1 and y = -2. Let's plug and chug:
-2 < 0.5(-1) + 2 -2 < -0.5 + 2 -2 < 1.5
Bingo! -2 is definitely less than 1.5. So, the point (-1, -2) is a solution.
4. The Point (-1, 2)
Now, let's test the point (-1, 2), where x = -1 and y = 2:
2 < 0.5(-1) + 2 2 < -0.5 + 2 2 < 1.5
Is 2 less than 1.5? Nope! 2 is greater than 1.5. Therefore, the point (-1, 2) is not a solution.
5. The Point (1, -2)
Last but not least, we have the point (1, -2), with x = 1 and y = -2. Time for the final substitution:
-2 < 0.5(1) + 2 -2 < 0.5 + 2 -2 < 2.5
Yes! -2 is less than 2.5. So, the point (1, -2) is a solution.
Putting It All Together: The Solutions Revealed
Phew! We've tested all five points. Now, let's recap our findings. We were asked to select three options that are solutions to the inequality y < 0.5x + 2. Based on our calculations, the solutions are:
- (-3, -2)
- (-1, -2)
- (1, -2)
Therefore, the three points that are solutions to the linear inequality y < 0.5x + 2 are (-3, -2), (-1, -2), and (1, -2).
Visualizing the Solution: A Graphical Approach
To really nail this down, let's think about what this looks like on a graph. The line y = 0.5x + 2 has a y-intercept of 2 and a slope of 0.5 (which means for every 1 unit you move to the right, you move up 0.5 units). Because our inequality is y < 0.5x + 2, we're interested in all the points that lie below this line. If you were to plot the five points we tested, you'd see that (-3, -2), (-1, -2), and (1, -2) all fall below the line, while (-2, 1) and (-1, 2) fall on or above the line. This visual confirmation is a great way to double-check your work and make sure you're on the right track.
Tips and Tricks for Solving Linear Inequalities
Now that we've conquered this problem, let's talk about some general tips and tricks for tackling linear inequalities:
- Understand the Inequality Sign: Pay close attention to whether you have <, >, ≤, or ≥. This will tell you whether the boundary line is included in the solution or not.
- Graphing is Your Friend: Visualizing the inequality on a graph can make it much easier to understand the solution set. Sketch the boundary line and shade the appropriate region.
- Test Points: When in doubt, test points! Choose a point in each region (above and below the line) and see if it satisfies the inequality. This will help you determine which region contains the solutions.
- Be Careful with Negatives: Remember that multiplying or dividing both sides of an inequality by a negative number flips the inequality sign. For example, if you have -2x < 4, dividing both sides by -2 gives you x > -2.
- Practice Makes Perfect: The more you work with linear inequalities, the more comfortable you'll become with them. Don't be afraid to try different problems and make mistakes – that's how you learn!
Real-World Applications of Linear Inequalities
You might be wondering,
