Inequality is a relationship in mathematics that compares two numbers or other mathematical expressions in a non-equal fashion. It's most typically used to compare the size of two numbers on a number line.
In other words, a linear inequality is a mathematical inequality that incorporates a linear function. One of the symbols of inequality is found in a linear inequality: In graph form, it depicts data that is not equal.
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Some of the linear inequality symbols are as follows:
- < less than
- > greater than
- ≤ less than or equal to
- ≥ greater than or equal to
- ≠ not equal to
- = equal to
Inequalities can be expressed as questions that are solved using similar procedures to equations, or as statements of fact in the form of theorems. It is used to compare numbers and find the range or ranges of values that satisfy a variable's criteria.
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Understanding Inequalities
Inequality can be understood with the help of an example:
| Symbol | Words | Example |
| > | Greater than | x + 3 > 4 |
| < | Less than | 6x < 24 |
| ≥ | Greater than equal to | 3y + 1 ≥7 |
| ≤ | Less than equal to | 4 ≤ x - 1 |
The below video is:
Linear Inequalities and their Solutions Detailed Video Explanation:
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Properties of Inequality
The properties of inequality can be understood by the following table:
Assume x, y, and z be real numbers
| Addition Property | If x is less than y, then x + z is less than y + z |
| Subtraction Property | If x is less than y, then x - z is less than y - z |
| Multiplication Property | If x is less than y, and z is greater than 0, then xz is greater than yz If x is less than y, and z is less than 0, then xz is greater than yz |
| Division Property | If x is less than y, and z greater than 0, then x÷z is less than y÷z If x is less than y, and z is less than 0, then x÷z is greater than y÷z |
| Transitive Property | If x is less than y, then y is less than z, then x is greater than z |
| Comparison Property | If x = y + z, and z greater than 0, then y is less than z, then x greater than y |
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Rules of Inequality
In order to perform of linear inequality operation one should follow the below rules:
- Suppose if a is less than b, then a + c is less than b + c. Adding the same number on both sides of inequality has no effect on the inequality symbol's direction.
- If a is less than b, then a-c is less than b-c . The orientation of an inequality sign is not changed by subtracting the same integer from each side.
- If a < b and if c is a positive number, then a × c < b × c.The orientation of an inequality sign is not changed by multiplying each side by a positive amount.
- If a < b and if c is a positive number, then ac < bc. Dividing each side of an inequality by a positive number has no effect on the inequality symbol's direction.
- Suppose if a < b and if c is a negative number, then a - c > b - c. The inequality sign is flipped when each side of the inequality is multiplied by a negative value.
- If a < b and if c is a negative number, then ac > bc. The direction of the inequality sign is reversed when either side of the inequality is divided by a negative number.
Process of solving linear inequality
With a few exceptions, inequalities can be solved using identical rules and methods as linear equations. When solving linear equations, the only difference is an operation that includes multiplication or division by a negative value. When you multiply or divide an inequality by a negative value, the inequality sign changes as mentioned in the above rules segment.
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The various operations applied while solving the linear inequality are as follows: Addition, subtraction, division, multiplication, and finally through distribution property.
Things to Remember
The following points should be always kept in mind while solving linear inequality:
- Remove fractions from the equation (multiplying all terms by the least common denominator of all fractions).
- Make things easier for yourself (combine like terms on each side of the inequality).
- Quantities can be increased or decreased (unknown on one side and the numbers on the other).
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Sample Questions
Ques: Solve the given inequality and give the solution in both inequalities?
4(m + 2) - 1 > 5 - 7 (4 - m)
Ans: 4m + 8 - 1 > 5 - 28 + 7m
= 4 m + 7 > -23 + 7m
= -3m > - 30
= m < 10
Hence the inequality from the solution is m < 10
Ques: Solve the given inequality and give the solution in both inequalities.
4z -1 + 2 < 10
Ans: In this, we will start off by subtracting 2 from all the parts, we get
= -3 < 4z < 8
Now dividing all by 4 we get,
= -¾ < z < 2
Hence the inequality from the solution is -¾ < x < 2
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Ques: Solve x + 3 < 5
Ans: solving the following equation we get,
= x < 5 - 3
= c < 2
Ques: Solve the inequality
1 < 2n + 3 less than 6
Ans: Now solving the following equation we get,
Now we subtract 3 from all the three sides
= -1+ (-3) < (2n + 3) + (-3) < 6 + (-3)
= -4 < 2n < 3
Dividing all sides by 2
= -2 < n < 3/2
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Ques: Solve the inequality
7x > 21
Ans: Now solving the following equation we get,
= x > 21/7
= x > 3
Ques: Solve the inequality
2x + 2 = 1
Ans: Now solving the following equation we get,
2x = 1 - 2
2x = -1
x = -½
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