Compound inequalities is a fundamental topic in mathematics where one could learn and use them in appropriate areas. To answer the question, “what is a compound inequality?” one should be able to understand what inequalities are. An inequality is nothing but the relationship between two or more non-equal expressions in mathematics. This is mostly used to compare numbers on the number line.
Therefore, a compound inequality can be explained as combining 2 or more simple inequalities in a mathematical expression using operations, namely OR & AND. To obtain a solution using an OR inequality, at least one of the values must be true, whereas both values must be true to obtain the result for AND operation.
What is compound inequality?
The solutions can lie between two quantities most of the time. It does not necessarily continue in a single direction endlessly. For instance, the systolic blood pressure range is 120 to 139 mm of mercury (Hg), which can be named borderline high blood pressure. This can easily be illustrated with the help of compound inequalities, b > 120 & b < 139. However, the other compound inequalities can be reunited by the term “OR”.
It has been described that the word “and” is used to join two inequalities. When these two are found to be true, then one can identify the occurrence of the solution of compound inequalities. This can be seen as a method in solving compound inequalities. In this, every equality tries to intersect or overlap with one another.
While joining two inequalities, and the function is OR, and either one of them is true, there will be compound inequality. So, one can clearly understand how to solve compound inequalities by knowing that the solution can be the combination or union of the provided two separate inequalities.
Solving compound inequalities in the “OR” form
Let us see how to solve compound inequalities that utilize OR to combine two separate inequalities. For instance, y > 6 OR y < 2, in this solution, it has been given that the value of y is either less than 2 or greater than 6. So, while graphing compound inequalities for this solution, one has to plot the graph of both the inequalities on the same line. The figure has been mentioned below:
One can see from the graph that it consists of an open circle at point 6 and a narrow blue line with an arrow mark pointing towards the right, and another open circle at point 2 with a red arrow pointing towards the left. Points 6 and 2 and all those in-between points in the graph are considered the only parts that cannot give results while solving compound inequalities. Apart from that, everything else can be a solution while graphing compound inequalities.
Consider a few more complex inequalities examples: Consider an example of OR compound inequality, y > 3 OR y < = 4. While graphing compound equalities, one can see an open circle on 3 while plotting y > 3. Here a blow arrow is being drawn, indicating the right direction. It demonstrates that all numbers that are greater than 3.
Meanwhile, there is a closed circle at 4 for the graph of y < = 4. Here, a red arrow towards the left is being used to indicate the numbers less than 4.
So, given below is the graphical representation of the same by combining both the graphs:
Compound inequalities in an OR statement can include all the numbers in every solution. In this case, it includes all the numbers contained on a number line. In the graph, the notation of a line that is greater than 3 & is less than or equal to 4 is illustrated in purple color. This is because it lies on both graphs. Therefore, one can know that the result of the compound inequalities y > 3 OR y < = 4 is nothing but the set of all the real numbers.
Let us see some more example problems to understand what compound inequality is complete.
Problem: Solve for y.
3 y-1<8 or y-5>0
y > 5 OR y < 3
This equation can either be solved by solving each inequality while isolating the variable or one can use the compound form using OR and can write both the inequality solutions.
Below mentioned is the graphical representation for the same:
One should remember applying the inequality properties while solving the compound inequalities. the following example carries a negative variable in order for it to isolate:
Problem: Solve for x.
2 x + 7 < 13 OR – 3 x – 2 < = 10
Answer: All real numbers. Reason being, x could be anything, less than 3 or greater than or equal to – 4.
In the above mentioned example, the inequalities have been solved separately in the first step. In addition to that, in the second step, the inequality sign has been reversed by a negative number while dividing.
Therefore, the solution set of x < 3 OR y > = 4 can be represented using the following graph:
Solving & Graphing compound inequalities in AND form
It is nothing but joining two individual inequalities with the help of the word AND to form one compound inequality. It can also be seen as the intersection of solutions to all the inequalities. In simple terms, both statements should be true in order to attain compound inequalities combining those statements.
The result of an AND statement can be determined by how common both inequalities are. If one has to think about it graphically, then it will be obtained as two graphs overlapping in their common spots.
While taking a look at the following compound inequalities example, y < 5 AND y > = – 1, the plotting of the graph and its every individual equality is shown like below:
Since it is clear that the operation AND joins two different inequalities, the solution can be found overlapping one another. Here, both statements are true at the same time.
Given below is the final result of the above-mentioned example:
While noticing from the graph, one can clearly rewrite y > = – 1 & y < 5 as – 1 <= y < 5. It is because the solution is found between – 1 and 5. It can be read as “y is greater than or equal to – 1 & is less than 5”. This sort of rewriting technique is only possible if the answer could be found between two numbers.
Let us see how to solve compound inequalities with the following examples using AND operation,
Problem: Solve for y.
1 – 4 y < = 21 AND 5 y + 2 > = 22
The graphs of these two inequalities are shown below in the form of a number line. Here, the result provided by the compound inequality is y > = 4, it is because this is the spot where both the graphs intersect.
The solution of these two graphs has been represented as:
y > = 4
Instead of splitting the compound inequalities into two inequalities and solving them, there is another efficient way to do it while also obtaining the same results. This is by applying the inequality properties to all three segments. For example:
Problem: Solve for y
3 < 2 y + 3 < = 7
0 < y < = 2
In this equation, the variable has been isolated by using the subtraction of 3 from all 3 parts of the inequalities, followed by dividing each part by 2.
In order for the learner to solve inequalities like a < y < b, the multiplication, and addition properties should be done to obtain the value for y.
Compound inequalities – Special cases
The result of a compound inequality using the AND operation is always considered as an overlap between two separate inequalities. It has been illustrated that there are three possible ways for it. They are:
- The solution can be of all the values between the two points.
- There is a possibility for a solution to start at one point and could be extended in one direction.
- If there is no overlap found between two inequalities, then there is no compound inequality solution.
From the above article, students could learn what compound inequality is, how to solve compound inequalities, and what compound inequality examples are. Furthermore, compound inequalities are easily manipulated and solved in the same way as any inequality problem because they represent the same as other simple inequalities: union or intersection operation.