Finding When Functions Are Positive
Hey guys! Let's dive into the world of functions and figure out when they turn positive. We've got four different functions to explore, and we'll break down each one step by step. It might sound a bit intimidating, but trust me, it's easier than it looks. So, grab your thinking caps, and let's get started!
1. Function y = 0.1x + 10
When tackling this function, y = 0.1x + 10, our mission is to discover the values of 'x' that make 'y' greater than zero. In simpler terms, we want to find out when the function's output is positive. This involves a bit of algebraic maneuvering, but don't worry, we'll take it slow and steady.
Setting Up the Inequality
First, we need to set up an inequality. We want to find when 0.1x + 10 > 0. This inequality represents the condition where the function's value is positive. It's the starting point for our journey to uncover the values of 'x' that satisfy this condition.
Solving for x
Next, we isolate 'x' on one side of the inequality. Subtract 10 from both sides: 0.1x > -10. Now, to get 'x' by itself, we divide both sides by 0.1: x > -100. Ta-da! We've found our solution.
Interpreting the Result
So, what does x > -100 actually mean? It means that for any value of 'x' greater than -100, the function y = 0.1x + 10 will be positive. Think of it like a number line – everything to the right of -100 makes the function happy and positive.
Visualizing the Function
Imagine a graph of this function. It's a straight line sloping upwards. It crosses the x-axis at -100. To the right of this point, the line is above the x-axis, indicating positive 'y' values. This visual representation can really help solidify your understanding.
Real-World Examples
Let's bring this to life with some examples. If x = -90, then y = 0.1(-90) + 10 = -9 + 10 = 1, which is positive. If x = 0, then y = 0.1(0) + 10 = 10, which is also positive. But if x = -110, then y = 0.1(-110) + 10 = -11 + 10 = -1, which is negative. See how it works?
Common Mistakes to Avoid
A common mistake is forgetting to flip the inequality sign when dividing by a negative number (which we didn't have to do here, thankfully!). Also, double-check your arithmetic to avoid simple calculation errors.
Wrapping Up
In summary, for the function y = 0.1x + 10, the function's value is positive when x > -100. We found this by setting up an inequality, solving for 'x', and interpreting the result. Keep practicing, and you'll become a pro at this in no time!
2. Function y = -0.1x + 10
Now, let's switch gears and analyze the function y = -0.1x + 10. This one's a bit different because of the negative sign in front of the 'x'. This negative sign will flip the script a little, so pay close attention.
Setting Up the Inequality
As before, we start by setting up the inequality to find when the function is positive: -0.1x + 10 > 0. Our goal is to isolate 'x' and see what values make this inequality true.
Solving for x
First, subtract 10 from both sides: -0.1x > -10. Now, here's the crucial part: we need to divide both sides by -0.1. Remember, when you divide (or multiply) an inequality by a negative number, you have to flip the inequality sign. So, we get: x < 100. That flip is super important!
Interpreting the Result
The solution x < 100 means that for any value of 'x' less than 100, the function y = -0.1x + 10 will be positive. Unlike the previous function, this one is positive for values to the left of 100 on the number line.
Visualizing the Function
The graph of this function is a straight line sloping downwards. It crosses the x-axis at 100. To the left of this point, the line is above the x-axis, indicating positive 'y' values. Notice how the negative sign changes the direction of the slope.
Real-World Examples
Let's plug in some numbers. If x = 90, then y = -0.1(90) + 10 = -9 + 10 = 1, which is positive. If x = 0, then y = -0.1(0) + 10 = 10, also positive. But if x = 110, then y = -0.1(110) + 10 = -11 + 10 = -1, which is negative. The trend is clear: values less than 100 make the function positive.
Common Mistakes to Avoid
The biggest mistake with this type of function is forgetting to flip the inequality sign when dividing by a negative number. Always double-check that step! Also, be careful with your arithmetic to prevent simple errors.
Wrapping Up
To recap, for the function y = -0.1x + 10, the function's value is positive when x < 100. The negative sign in front of 'x' made all the difference, so keep an eye out for those sneaky negatives!
3. Function y = -0.1x - 10
Alright, let's keep the ball rolling with the function y = -0.1x - 10. This one has a negative 'x' term and a negative constant, which will give us a different range of positive values.
Setting Up the Inequality
As always, we start by setting up the inequality to find when the function is positive: -0.1x - 10 > 0. We want to find the 'x' values that make this true.
Solving for x
First, add 10 to both sides: -0.1x > 10. Now, we divide both sides by -0.1, and don't forget to flip the inequality sign: x < -100. This flip is crucial because we're dividing by a negative number.
Interpreting the Result
The solution x < -100 means that for any value of 'x' less than -100, the function y = -0.1x - 10 will be positive. So, values to the left of -100 on the number line make this function positive.
Visualizing the Function
The graph of this function is a straight line sloping downwards. It crosses the x-axis at -100. To the left of this point, the line is above the x-axis, indicating positive 'y' values.
Real-World Examples
Let's test some values. If x = -110, then y = -0.1(-110) - 10 = 11 - 10 = 1, which is positive. If x = -200, then y = -0.1(-200) - 10 = 20 - 10 = 10, also positive. But if x = -90, then y = -0.1(-90) - 10 = 9 - 10 = -1, which is negative.
Common Mistakes to Avoid
The most common mistake, once again, is forgetting to flip the inequality sign when dividing by a negative number. Always double-check that step! Also, watch out for arithmetic errors.
Wrapping Up
In summary, for the function y = -0.1x - 10, the function's value is positive when x < -100. The combination of negative 'x' and a negative constant shifted the positive region to the left of -100.
4. Function y = 0.1x - 10
Last but not least, let's tackle the function y = 0.1x - 10. This one has a positive 'x' term and a negative constant. Let's see what values make it positive.
Setting Up the Inequality
We set up the inequality: 0.1x - 10 > 0. We need to find the 'x' values that make this true.
Solving for x
First, add 10 to both sides: 0.1x > 10. Now, divide both sides by 0.1: x > 100. No need to flip the inequality sign this time since we're dividing by a positive number.
Interpreting the Result
The solution x > 100 means that for any value of 'x' greater than 100, the function y = 0.1x - 10 will be positive. So, values to the right of 100 on the number line make this function positive.
Visualizing the Function
The graph of this function is a straight line sloping upwards. It crosses the x-axis at 100. To the right of this point, the line is above the x-axis, indicating positive 'y' values.
Real-World Examples
Let's plug in some values. If x = 110, then y = 0.1(110) - 10 = 11 - 10 = 1, which is positive. If x = 200, then y = 0.1(200) - 10 = 20 - 10 = 10, also positive. But if x = 90, then y = 0.1(90) - 10 = 9 - 10 = -1, which is negative.
Common Mistakes to Avoid
Make sure to double-check your arithmetic and ensure you're adding and dividing correctly. There's no tricky sign flipping here, so it's a bit more straightforward.
Wrapping Up
In summary, for the function y = 0.1x - 10, the function's value is positive when x > 100. The positive 'x' and negative constant combination shifted the positive region to the right of 100.
Final Thoughts
So, there you have it! We've explored four different functions and found the values of 'x' that make each one positive. Remember, the key is to set up the inequality correctly, solve for 'x', and pay close attention to those negative signs! Keep practicing, and you'll become a master of finding positive function values in no time. Good luck, and happy solving!