Finding the interval of values a variable takes on is important in many situations. For instance, if you are given the graph of a function f(x) = x + 4, you can find the range of values x takes on by finding where the graph equals 0.
You then can find all of the possible values x can take on by using either the left or right boundary of where x = 0 as this is an inclusive range.
This article will discuss how to find an interval of numbers that a variable takes on given a graph. This can be applied to many situations such as solving problems with linear equations or finding possible values of a variable based on its graph.
Before getting into how to find an interval of numbers that a variable takes on, let’s first go over what a domain and range are and how to find them.
Find where the function is negative
The first step is to find the minimum value of the function. This is where the function is zero, or where it has a descending curve. In this case, the minimum value of the function is four, since that is where the X value of four sits and there is no lower curve.
Then, you must find a number Δ such that if |X − 4| > Δ then X − 2 > 0.4. Using algebra, you can write this as:
If |X − 4| > 0.4 Then X − 2 > 0.2
This means if X is any number greater than 0.4 then X − 2 is greater than 0.2. For example, if X = 0.6 then (X−2) = 0.4 and so (0.6−0.4) = 0.2, which is greater than 0 .2.
Use the half-interval method
The other method to find a number Δ such that if |X − 4| 0.4 is to use the half-interval method. This method uses two intervals that have an upper limit of 0.4 on the Y-axis and an interval of -2 on the X-axis with 4 as the middle point.
The first interval, from -2 to 0, includes all numbers greater than -2 and equal to 0. The second interval, from 0 to 4, includes all numbers greater than 0 and equal to 4. These two intervals are crossed out on the graph provided in the article.
Then, any number between -2 and 0 that is greater than or equal to -2 and less than or equal to 0 is included in the first interval. Any number between 0 and 4 that is greater than or equal to 0 and less than or equal to 4 is included in the second interval.
Use the difference of squares method
The last method requires you to first find the zeroes of the derivative. Once you find the zeroes, you can find the number Δ using the Given Graph of F(x) = X to Find a Number Δ Such That if |X − 4| ≪ Δ Then X − 2 ≪ 0.4 graph.
Zero is found at x = 2, so write down 2 as your first zero. Next, find all of the points where f’(x) = 0 by checking each point on the graph and doing a little math. There are three points where this is true: (2,0), (−1,0), and (0,−1).
Add these up and divide by two to get one zero left over. Put this zero in between the two already there to create a Δ.
Check your answer using a calculator
In this case, your calculator will tell you that the answer is 2. Therefore, the correct answer is X = 2.
Calculators can be very helpful in solving these problems. Make sure that you are checking your answers with a calculator to ensure that it is correct!
Closing Thoughts
Solving inequalities is an important math concept that ties into many other areas of mathematics and science. As you progress in your education and career, you will encounter problems that require you to solve inequalities.
You should now be able to identify linear inequalities, find solutions to arithmetic inequalities, and solve inequality calculators using logic and steps outlined in this article.
Interpret the result
As shown in the given graph, if |X − 4| is less than Δ, then X − 2 is less than 0.4. In other words, there is a number X such that if the difference between X and 4 is less than Δ, then X − 2 is less than 0.4.
This could be interpreted as if there is a number that increases by 2, then the new value will be greater than 0.4 greater than 4.
This problem could be applied to many situations. For example, it could be applied to traffic laws where the given speed limit is enforced with a device that detects whether or not you are within a certain percentage of the limit. If you are within 0.4 of the limit, you are legally driving at the limit.
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