Finding Positive Values: A Guide To Linear Functions

Hey guys! Let's dive into a common algebra problem: determining the values of x for which linear functions are positive. This is super useful, whether you're working on homework, trying to understand how things change in the real world, or just brushing up on your math skills. We'll break down each function step-by-step, making sure it's crystal clear. We'll explore four different linear equations: y = 0.1x - 10, y = -0.1x + 10, y = -0.1x - 10, and y = 0.1x + 10. Our goal is to figure out the range of x values that will result in a positive y value for each equation. Ready? Let's get started!

Understanding the Basics: Linear Functions and Positivity

Alright, before we get our hands dirty with the specific equations, let's quickly recap what makes a linear function and what it means for a function to be positive. Linear functions, in the form of y = mx + b, are straight lines on a graph. The 'm' represents the slope (how steep the line is), and 'b' represents the y-intercept (where the line crosses the y-axis). When we say a function is positive, we mean that the y value is greater than zero (y > 0). Graphically, this is the region above the x-axis. So, our main task is to find the x values that place our function's line above the x-axis. This involves a bit of algebra, a little bit of intuition about how lines behave, and a whole lot of fun (well, maybe not a whole lot, but definitely some!).

To figure out where the function is positive, we'll generally follow these steps: First, we set the function equal to zero (y = 0) to find the x-intercept, which is the point where the line crosses the x-axis. Then, we can use this intercept and the slope to determine the intervals where the function is positive (above the x-axis) and negative (below the x-axis). The slope tells us whether the line is going up or down as we move from left to right. This helps us decide which side of the x-intercept the function will be positive. So, with this basic understanding, let's start working through each equation, shall we?

We need to find the critical point (where y = 0), and then we'll test the intervals to see where y is positive. The x-intercept is crucial because it divides the number line into two parts: one where the function is positive, and another where it's negative. The sign of the slope will tell us whether the function increases or decreases to the right of the x-intercept. A positive slope means the function goes up as x increases, and a negative slope means it goes down. Keep these concepts in mind as we analyze each equation. Remember, it's all about finding the x values that make y greater than zero. Let’s get into the specifics!

Analyzing Each Function

1. y = 0.1x - 10

Let's start with our first function, y = 0.1x - 10. First, we need to find the x-intercept. We do this by setting y = 0: 0 = 0.1x - 10. Now, let's solve for x. Add 10 to both sides: 10 = 0.1x. Then, divide both sides by 0.1: x = 100. So, the x-intercept is at x = 100. This means the line crosses the x-axis at the point (100, 0). The slope of this line is 0.1, which is positive. This means that as x increases, y also increases, and the line slopes upwards from left to right. Thus, the function is positive to the right of the x-intercept (where x > 100). To verify, pick a value greater than 100, say 150, and plug it into the equation: y = 0.1(150) - 10 = 15 - 10 = 5. Since 5 is positive, our analysis is correct. The solution is x > 100. This is pretty straightforward, right?

2. y = -0.1x + 10

Next up, we have y = -0.1x + 10. Again, start by finding the x-intercept. Set y = 0: 0 = -0.1x + 10. Subtract 10 from both sides: -10 = -0.1x. Divide both sides by -0.1: x = 100. So, the x-intercept is again at x = 100. However, the slope here is -0.1, which is negative. This means that as x increases, y decreases, and the line slopes downwards from left to right. Therefore, the function is positive to the left of the x-intercept (where x < 100). Let’s double-check. Choose a value less than 100, let's try 50: y = -0.1(50) + 10 = -5 + 10 = 5. As you can see, 5 is indeed positive. So, the solution is x < 100. Always double-checking our work is a good habit.

3. y = -0.1x - 10

Alright, let's look at y = -0.1x - 10. Set y = 0: 0 = -0.1x - 10. Add 10 to both sides: 10 = -0.1x. Divide both sides by -0.1: x = -100. The x-intercept is at x = -100. The slope here is -0.1, which is negative. Because the slope is negative, the line slopes downward. The function will be positive to the left of the x-intercept. But let's check. Take a value less than -100, say -150: y = -0.1(-150) - 10 = 15 - 10 = 5. As you can see, 5 is indeed positive, but the x value is less than -100. This is the first time we've encountered a negative x-intercept, which requires a little bit more attention. So, the solution is x < -100. Remember, a negative slope means the line is decreasing as we move from left to right, and the x-intercept helps us identify the interval of positive values. Keep an eye on those negative signs and make sure you're careful when working through each step!

4. y = 0.1x + 10

Lastly, we'll examine y = 0.1x + 10. Set y = 0: 0 = 0.1x + 10. Subtract 10 from both sides: -10 = 0.1x. Divide by 0.1: x = -100. The x-intercept is at x = -100. This time, the slope is 0.1, which is positive. The line slopes upward. Therefore, the function is positive to the right of the x-intercept (where x > -100). Let's test it out. Choose a value greater than -100, for instance, 0: y = 0.1(0) + 10 = 0 + 10 = 10. Ten is positive. This means our solution is x > -100. Now you've worked through the final equation.

Conclusion: Summarizing the Results and Tips for Success

So, there you have it, guys! We have successfully determined the intervals where each of our functions is positive. Let's summarize our findings:

• For y = 0.1x - 10, the function is positive when x > 100.
• For y = -0.1x + 10, the function is positive when x < 100.
• For y = -0.1x - 10, the function is positive when x < -100.
• For y = 0.1x + 10, the function is positive when x > -100.

Important Tips:

• Always Find the x-intercept: This is your key to splitting the number line into regions. Set y = 0 and solve for x.
• Understand the Slope: A positive slope means the line goes up from left to right; a negative slope means it goes down.
• Test a Value: After finding the x-intercept, pick a number on either side of it and plug it into the original equation to check if the result is positive. This helps avoid errors. Also, use a calculator to help make sure you don't mess up the arithmetic.
• Pay Attention to Signs: Keep careful track of positive and negative signs. They can change the solution's direction. Take your time and go step by step.
• Practice Regularly: The more you work through these types of problems, the easier they become. Practice makes perfect. So, keep practicing until it becomes second nature.

I hope this guide has been useful in understanding how to find the intervals of positivity for linear functions. Remember to take things one step at a time, and don’t be afraid to double-check your work. You've got this! And if you get stuck, don't hesitate to go back through the steps. Keep practicing, and you’ll master it in no time! Good luck with your algebra adventures, and keep exploring the amazing world of mathematics!