The interval is – In Mathematics there is material that discusses the comparison of the sizes of two or more things, namely inequality. To carry out its function, an inequality requires several signs such as < (less than), < (less than or equal to), > (more than), > (more than or equal to).
For example, you want to say that the value of x is “less than” a, then it can be written as “x<a”. Or you want to say that the value of x is “more than or equal to” a, if written it will be “x>a”.
So, to understand inequality material, you have to know the properties of inequality, intervals, and ways to solve inequalities. In this article, we will focus on intervals by alluding a little to the nature of inequalities. Listen to the end, OK!
Properties of Inequality
Inequalities in mathematics have certain properties, including:
Nature 1
“The sign of the inequality will never change if you subtract or add it with other numbers or mathematical expressions.”
For example, if there is an inequality that says a > b then this means:
a + c > b + c and a – c > b – c
But if the inequality is a < b, then it means:
a + c < b + c and a – c < b – c To make it more understandable, let’s try to enter a real example like this: x + 6 > 8 -> x + 6 – 6 > 8 – 6 -> x > 2
Nature 2
“The sign of the inequality also won’t change if you divide or multiply it by a positive number.”
For example, there is an inequality which states that a > b and c > 0, it means:
ac > bc and a/c > b/c
For example, an inequality 4x > 12 if each side is divided by 4 (positive) then the result will be like this:
4x/4 ≥ 12/ 4 -> x ≥ 3
Nature 3
“The sign of the inequality is reversed when divided or multiplied by a negative number.”
If the inequality a > b and c < 0, it means ac < bc and a/c < b/c. Be careful, quite a few people forget that they have to flip signs after multiplying or dividing an inequality. For example like this: -3x > 9 can only be solved by dividing the right and left sides by the number -3 or you can also multiply each side by the number -⅓. Why so? Yes because if the inequality is multiplied by a negative number, then the sign will be reversed.
If written in full it would be like this: -3x > 9 -> 3x/-3 < 9/3 -> x < -3. Property 4 “If the inequality is raised to a power, then the sign can be reversed or not, depending on the odd or even rank.” For example, if there is an inequality a > b > 0, this means:
a2 > b2 > 0
a3 > b3 > 0
a4 > b4 > 0
a5 > b5 > 0
and so on. Generally an > n if a is a natural number
If a < b < 0, then it becomes: a2 > b2 > 0
a3 < b3 < 0 a4 > b4 > 0
a5 < b5 < 0 And so on. Generally, an > bn if n is even and an < bn if n is odd.
Definition of Intervals
When studying inequalities you will definitely find a line of real numbers in it. This is a one-to-one combination between the points in the horizontal straight line and the numbers in the real set.
Simply put, the number line is a ruler for placing each real number in order based on its value. Like the example below:
In the picture you can see that you can know or estimate the position of each number on a straight line. For example the integers (0, 2, and 4) occupy their respective places where 0 is to the left of 2 and 2 to the left of 4.
Or the rational number (4/5) whose position is on the left–almost close to–1 because the value of 4/5 is equal to 0.8. Likewise with the irrational number (root 2) which is to the left of the number 2 because its value is 1.4.
Every real number has a position on the number line and you can determine the position of a number by looking at the positions of other numbers that are close in value.
Now, in this number line there is something called an interval or a set of real numbers that is limited by one or two number boundaries. For example, like 2 < x < 5. This set of real numbers can be understood as the set of numbers whose value is greater than 2 but less than 5.
This means that there are numbers 3, 4, 8/3, 10, -5, and ½ which are part of the set. The numbers 3, 4, and 8/3 are numbers that enter the set and are called interior points. While the numbers 10, -5, ½ are outside the set and are called exterior points.
From this explanation it can be concluded that the interval is a set that is part of the real number line which has at least 2 different real numbers and all the real numbers that are in between.
Types of Intervals
Generally the interval is divided into several types, namely:
1. Open Intervals
This is an interval that does not include the extreme points to which it is included but includes all values that fall between them. Open intervals are represented by:
a < x < b or (a;b).
For example 2 < x < 6 can be written (2,6) which means the set of all real numbers whose values are greater than 2 and less than 6. This means that 2 and 6 are interval boundaries that are outside the interval itself (exterior point). If depicted on a number line it will be like this:
The sign in numbers 2 and 6 is a circle without content because they do not enter into the set of real numbers in that interval.
2. Closed intervals
This is an interval type that includes the extremes of the interval as well as all values in between. Closed intervals are represented by: a ≤ x ≤ b or [a; b].
For example 0 < x < 72 can be written as [0, 72] which means the set of real numbers with values greater than or equal to 0 and less than or equal to 72.
The difference between closed intervals and open intervals lies in the boundary of the interval that enters the interior point. If depicted on a number line it would be like this:
The interval boundaries 0 and 72 are circles that have this because these boundaries enter into the subset of the set or interior points.
3. Semi Open Intervals
The third type is an interval that only includes one of the extreme values between the two, so that the other extreme values are excluded. Including right end and left end, can be included or can be excluded. Semi-open intervals are represented by a ≤ x < b or a < x ≤ b
For example, you have a semi-open interval (1, 5], so you will have a set of numbers whose value is greater than 1 and less than or equal to 5, but 1 is not included, yes. If you describe it more or less like this:
4. Infinite intervals
Infinite intervals are often referred to as infinite intervals. This is a type of interval that has an infinite value at either or both ends where the infinite end will be the open end. Meanwhile, if both ends have infinity then it will become a real line.
Infinite intervals are represented by a ≤ x or x ≤ a, which will be [a, ∞) or (-∞, a). Or it can also be written like this: [a; ∞).
For example you have an infinite interval [1; ∞), then you will have a set of numbers whose value is greater than or equal to 1 and so on until infinity. If described more or less like this:
5. Limited Intervals
The last type, namely bounded intervals, has two other categories, namely right bounded intervals and left bounded intervals. So, right and left here mean the maximum and minimum of each number.
For example, a right-open interval means no maximum and a left-open interval has no minimum.
How to Write Intervals
The limits that an interval has are usually indicated by “brackets”. For semi-open intervals and open intervals, square brackets are used. For example like [a ; b] for open intervals and [a ; b and a ; b] for semi-open intervals.
When writing intervals, the intervals are always written in ascending order and never in descending order. If the parentheses are closed, any boundary points are included while if the parentheses are open, the endpoints are omitted.
For example, the set of integers { 0 ; 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9}. This set contains all integers from 0 to 9 inclusively.
However, not all sets are written using integers like that, there are also abstract sets which are written using letters and these are usually found in algebra materials.
Some letters that are often used in abstract sets are:
You’ll also find math marks tasked with indicating the two sets of real numbers and the interactions they have. At first, these signs can seem very complicated and abstract, but with enough practice, they can seem simple and easier to work with.
Apart from showing two sets of real numbers, these mathematical signs can also be used to visualize mathematical data which is usually used when you want to see the interactions that exist in an interval.
Here are the most frequently used signs:
Intervals Example
To make it easier for you to understand the concept of intervals, here are examples of intervals, classifications, and numbers entered.
Ways to Complete Intervals
You can complete an interval by reading and planning it and then seeing what will be included and what will not be included. But remember, pay attention to the direction of the symbols “> (more than)” and “< (less than)” because they determine the end point of an interval. For example like this:
Closed Intervals
A closed interval is an interval that has an endpoint. Thus you can use two different ways when there are two real numbers that interact with x.
[a ; b] = a ≤ x ≤ b [a ; b] = a ≤ x < b
]a ; b] = a < x ≤ b
]a ; b[= a < x < b If the brackets are closed, it means that x is greater than or equal to b.
Meanwhile, if the brackets are opened, it means that x is more than and less than b. Open Interval Open interval is the interval in which a and b are two different numbers.
[a ; ∞[= x ≥ a ]a ; ∞[= x > a
]- ∞ ; b] = x ≤ b
]- ∞ ; b[= x < b
Well, this open interval doesn’t include the endpoints so you won’t know where the endpoints are.
Types of Interaction Between Intervals
The slice that exists between the interval [a ; b] with [c ; d] is a set of real numbers x located at [a ; b] and [c ; d]. This condition is denoted by the symbol ∩.
Try to imagine that a, b, c, and d are integers with intersection I like this: