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Ellipse: Definition, Properties, Applications, Equation, Formulas
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Ellipse: Definition, Properties, Applications, Equation, Formulas
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April 8, 2025Combining Inequalities in One Variable: A combining inequality is a sentence with two inequality statements connected by the words “or” or “and.” The word “and” denotes that both compound sentences’ statements are true at the same time. It’s the intersection or overlap of the solution sets for each statement. The word “or” signifies that the entire compound sentence is true if either statement is true.
A conjunction is a compound inequality that contains the word “and.” Although the words “and” and “or” are both conjunctions in speech, the mathematical conjunction has a distinct meaning than the grammatical one. To demonstrate the idea, the conjunction (part of speech) “or” generates a disjunction when used in a compound inequality. Remember that “con” implies “with another “and” “dis” indicates “one OR the other.”
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Linear inequalities are defined as expressions that compare two values using inequality symbols.
We know the larger (greater) or smaller (lesser) numbers for each other or any third amount. Only three conditions hold between any two quantities (say
Addition Property of Inequality Definition
The addition property of linear inequality states that adding the same number to each side of the inequality produces an equivalent inequality, in which the inequality symbol remains the same.
Example:
Subtraction Property of Inequality Definition
The linear inequality subtraction property states that removing the same number from both sides results in an identical inequality, i.e., the inequality symbol remains the same. Example:
Multiplication Property of Inequality Definition
Multiplying both sides of an inequality by a positive number always results in an equivalent inequality, i.e., the inequality symbol remains the same, according to the linear inequality multiplication property.
Example:
However, the inequality symbol is reversed if multiplication is done with a negative value on both sides of the inequality.
Example:
Multiply the given inequation by
Thus,
Division Property of Inequality Definition
The division property of linear inequality states that dividing both sides of an inequality by a positive number produces an analogous inequality, i.e., the inequality symbol remains the same.
Example:
Divide the given inequation by
Thus,
However, the inequality sign is reversed if we divide both sides of an inequality with a negative number.
Example:
Divide the given inequation by
Thus,
Learn Properties of Inequalities
If one side of the inequality is multiplied by a negative number, the inequality’s “direction” must be changed to get an equivalent inequality (for example,
That is, if
1.
2.
When the word “and” is used to unite two inequalities, the compound inequality is solved when both inequalities are true simultaneously. It is the intersection, or overlap, of the solutions to each inequality. When the two inequalities are connected by the word or, the compound inequality is solved when one of the inequalities is true.
A compound inequality comprises at least two inequalities connected by “and” or “or.” The combination, or union, of the two distinct solutions, is the solution.
Example: Consider the inequality
In other words,
The intersection of the two inequalities
For example, the inequality
We may describe each of these inequalities separately using interval notation:
The applications of linear inequalities are:
Learn Concepts on Linear Inequalities
Q.1. Solve for
Ans: By solving the first inequality for
By solving the second inequality for
In terms of graphs, we have:
As a result, we may represent our compound inequality as a simple inequality:
Q.2. Solve for
Ans: By solving the first inequality for
By solving the second inequality for
In terms of graphs, we have:
From the above graph, we interpret that there are no solutions to the compound inequality because there is no value of
Q.3. Solve for
Ans: By solving the first inequality for
By solving the second inequality for
All the numbers to the left of
{
Q.4. Solve for
Ans: By solving each inequality separately, we get
To isolate the variable from the first equation, divide both sides by two, obtaining;
Now, by subtracting
Hence, the compound inequality is
Q.5. Solve:
Ans: By solving each inequality separately, we get
Consider the inequation
So,
Consider the inequation
So,
Hence, the solution to this combined inequation is
In this article, we learnt about linear inequalities, properties of inequalities (addition, subtraction, multiplication, and division property of inequalities), equivalent inequality, compound inequalities, applications of linear inequalities, solved examples on combining inequalities in one variable, FAQs on combining inequalities in one variable.
This article’s learning outcome is used to find the maximum, and minimum values of a situation with many restrictions are frequently determined using a system of linear inequalities.
Q.1. How do you combine two inequalities?
Ans: When the word “and” is used to unite two inequalities, the compound inequality is solved when both inequalities are true simultaneously. It is the intersection, or overlap, of the solutions to each inequality. When the two inequalities are connected by the word or, the compound inequality is solved when one of the inequalities is true.
Example: Solve:
Solution: By solving each inequality separately, we get
Therefore, the solution is
Hence, the combined inequation is
Q.2. How do you solve an inequality with one variable?
Ans: There are the following steps used to solve the linear inequation in one variable.
1. Remove the fractions (or decimals) by multiplying both sides by an appropriate factor.
2. Put all variable terms on one side and all constants on the other side.
3. Make the coefficient of the variable
4. Choose the solution set from the replacement set.
Q.3. What is inequality with one variable?
Ans: If and only if the exponent of each variable in the inequation is one, it is said to be linear (or each variable appears in first degree only). No word involves the variables’ product.
Example:
Q.4. What are the two types of compound inequalities?
Ans: In other circumstances, inequality can be used to express more than one restricting value. In these cases, a compound inequality is used.
As a result, a compound inequality can be defined as an expression having two inequality statements connected by the terms “AND” or “OR.”
The conjunction “and” denotes the truth of two statements at the same time.
On the other hand, the term “or” denotes that the entire compound statement is true if one of the statements is true.
A combination of the solution sets for the individual statements is denoted by the term “or.”
Q.5. What is the difference between properties of equality and properties of inequality?
Ans: The solution set will not change if you use the properties of equality on any equation. The inequality symbol must be reversed when multiplying or dividing an inequality by a negative number; otherwise, the solution set will be incorrect.
Now you are provided with all the necessary information on combining inequalities in one variable and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.
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