Laws of Inequality

Laws of Inequality – Definition, Meaning, Facts, Examples | Rules for Switching Inequality Signs

This entire article deals with the law of Inequality. In maths, inequality will occur when two mathematical statements or two numbers are compared in a non-equal way. Generally, inequalities can be either numerical or algebraic in nature or a combination of the two numbers. When two linear algebraic expressions of degree 1 are compared, linear inequalities occur.

You can learn about the definition of laws of inequality, properties, solved example problems on the law of inequality, and so on. Students who are lagging in inequality topics can read our articles and practice well for the exams. So, without being late or waiting a minute, let’s begin your practice by referring to our 10th-grade math concepts.

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Properties of Inequality Definition

Laws of inequality are defined as if you add the same number to both sides of an inequality, the inequality remains true. Suppose, if you subtract the same number from both sides of the inequality, the inequality remains true, same as if you multiply or divide both sides of an inequality by the same positive number, the inequality will remain true.

So, an inequality’s solutions are all x values that make the inequality true. The solution is usually a set of x values that we plot on a number line.

Properties of Inequality | Laws of Inequality Rules PDF

An inequation is said to be linear if and only if the exponent of each variable in it is one means each variable occurs in the first degree only and there is no term involving the variables product. The following are the properties of laws of inequality:

Property 1(Addition and Subtraction Property): If the same quantity is added to or subtracted from both sides of an inequation, then the sign of inequality between the two sides does not change. Symbolically, we can express this as follows,
If, a > b then a + c > b + c is for addition, and a – c > b – c is for subtraction.
Examples: (i) If x – 7 > 5 then x – 7 + 7 > 5 + 7 or x > 12.
(ii) If y – 5 > 11 then y – 5 + 5 > 11 + 5 or y > 16.

Property 2(Multiplication Property): If both sides of an inequation are multiplied by the same positive quantity, the symbol of inequality does not change. Symbolically, it can be expressed as follows. It will hold for all inequalities.
If a < b then ac < bc where ‘c’ is a positive integer or quantity.
Example: If x<2, then it will be x×3 < 2×3.
Multiplication with a negative number on both sides of the inequality, on the other hand, does not yield an identical inequality unless the inequality symbol is also reversed.
Example: If x<2
Multiply the given inequation by −2.
Thus, it will be −2x>−4.

Property 3(Division Property): According to the division property of linear inequality, dividing both sides of an inequality with a positive number results in an equivalent inequality, i.e., the inequality symbol remains the same.
Example: If 2x<4
Divide the given inequation by 2.
Thus, 2x/2< 4/2.
However, if the inequality sign is reversed, dividing both sides of an inequality with a negative number gives an equivalent inequality.
Example: If −3x>9
Divide the given inequation by −3.
Thus, it will be −3x/−3 < 9/-3.

Property 4 (Transitive Property): In this, the term “transitive” will refer to the word transfer. The transitive property states that if a,b, and c are three quantities, and if a is connected to b by some rule and b is related to c by the same rule, then a and c are related to each other by the same rule.”
If a<b and b<c then a<c.
Example: Suppose 10 > 5 and 5 > 2, then 10 > 2
that is in the form of a<b and b< c, then a < c.
So, it will be 5 < 10 and 10 < 20, then 5 < 20.

Property 5(Comparison Property): The Comparison property in mathematics in inequality is a relation that compares two numbers or other mathematical expressions in a non-equal way. It’s mostly used to compare two numbers on a number line based on their size. It will be in the form of,
If a=b+c and c>0, then a>b
Example: 8=5+3, then 8>5.

The table below given is for summarizes the properties of Inequality,

Laws of Inequality Math Problems with Solutions

Problem 1: Show that the sign of inequality remains the same if we add and subtract 3 and 2 respectively from the following inequalities
(i) 7<10
(ii) 5>−7

Solution: 
As given in the question,
(i) 7<10
First, by adding 3 on both sides, we get
7+3 < 10+3
i.e., 10<13
Next, subtract 2 from both sides.
10−2 < 13−2
then the value is 8<11.?
So, it is true.
Therefore, the sign of the given inequality remains the same.

(ii) Given as −5>−7
By adding 3 on both sides. we get the value is
−5+3 > −7+3 = −2>−4
Now, subtract 2 from both sides.
−2−2 > −4−2
then the value is −4>−6,
Hence, it is true.
Therefore, the sign of the given inequality remains the same.

Problem 2: Calculate the range of values of x, which satisfies the inequality: 2x − 8 < 4x + 10.

Solution:
In the given question, the inequation is 2x − 8 < 4x + 10 where x is a variable.
Now, we need to find the range of the x value.
So, first, add 8 on both sides of the above equation. We get,
2x − 8 + 8 < 4x + 10 + 8
then it is 2x < 4x + 18.
Next, subtract 4x on both sides of the above equation. Then it will be
2x – 4x < 4x – 4x + 18
-2x < 18.
Next, we will divide the above equation with -2 into both sides. Then the inequality reverses by multiplying or dividing both sides by -1.
-2x/-2 < 18/-2
So, it will be x > -9
The value of x will be greater than -9.
Therefore, the value of x is greater than -9 i.e., x>-9.

Problem 3: Mark the statement as true or false. Justify your answer.
(i) If 2x + 12 > 30 then 2x – 12 > 6.
(ii) If 8m > – 48 then – 2m > 12.

Solution: 
As given that,
(i) Given equation is 2x + 12 > 30 where x is the variable.
2x + 12 > 30
First subtract 24 on both sides of the given equation.
2x + 12 – 24 > 30 – 24
then it will be 2x – 12 > 6
Therefore, the given statement if 2x + 12 > 30 then 2x – 12 > 6 is true.

(ii) Given the equation is 8m > – 48, where m is the variable.
8m > – 48
Now, divide the above equation with -4 on both sides. So, the inequality reverses on multiplying or dividing both sides by -1.
8m/(-4) > – 48/-4
-2m < 12
Hence, the given statement 8m > – 48 then – 2m > 12 will be false.

Problem 4: Find the values of x from the given inequations.
(i) – 2x ≥ -3x + 5, find the least positive value of x.
(ii) 4x/3≤ 2x/ 6  + 2, find the value of x.
(iii) 4x + 8 ≥ 8x, what is the highest positive number of x?

Solution: 
In the given question,
(i) Given that – 2x ≥ -3x + 5
– 2x ≥ -3x + 5
Now, subtracting 5 on both sides of the given equation.
– 2x – 5 ≥ -3x – 5 + 5
– 2x – 5 ≥ -3x
Next, subtract -2x to the right side of the above equation.
-5 ≥ -3x + 2x
-5 ≥ -x
Now, multiply the above equation with -1 on both sides.
5 ≤ x
Therefore, the least positive value of x is 1.

(ii) Given that 4x/3 ≤ 2x/6 + 2
Now, we will find the value of the x.
First, multiply the above equation with 3 on both sides.
34x/3 ≤ 32x/6 + 2 (3)
4x ≤ 2x/2 + 6
4x ≤ x + 6
Now, move the x to the left side of the above equation. We get,
4x – x ≤ 6
3x ≤ 6
Now, divide the above equation by 3 on both sides.
3x/3 ≤ 6/3
then it will be x ≤ 2.
Therefore, the value of x is 2.

(iii) Given that 4x + 8 ≥ 8x
4x + 8 ≥ 8x
First, move 4x to the right side of the above equation.
8 ≥ 8x – 4x
8 ≥ 4x
Now, divide the above equation with 4 on both sides.
8/4 ≥ 4x/4
2 ≥ x
Therefore, the highest positive number of x is 2.

Problem 5: An integer is such that one-third of the following integer is at least 2 more than one-fourth of the previous integer. Find the smallest value of the integer.

Solution: 
In the given question,
Let the x be an integer, then one-third of the following integer is x+13, and one-fourth of the previous integer is x–14.
According to the question, it will be x+13 ≥ x–14+2
Now, multiply both sides by 12. We get
12(x+1)3 ≥12(x–1)4+2×12
i.e., 4(x+1)≥3(x–1)+24
4x+4 ≥ 3x−3+24
Now, subtract (3x+4) from both sides, then it will be
4x+4–(3x+4) ≥ 3x+21–(3x+4)
4x+4–3x–4≥3x+21–3x–4
Then the value is x≥17
Therefore, the smallest value of x is 17.

Explore our article on Worksheet on Laws of Inequality as well to practice more such problems and get a grip of the concept.

FAQ’s on Laws of Inequality

1. What are the types of inequalities?

There are four types of inequalities, namely:
(i) Less than – <
(ii) Greater than – <
(iii) Less than or equal – ≤
(iv) Greater than or equal – ≥

2. Mention the rules for solving inequalities?

The following rules do not affect the direction of inequality:
(i) Add or subtract the same number on both sides of the inequality.
(Ii) Multiply or divide the same positive number on both sides of the inequality.
(iii) Simplify a side of the equation if possible.

3. Does swapping the values of the left and right-hand sides change the direction of inequality?

Yes, swapping the values of the left and right-hand sides changes the direction of the inequality.

4. What is meant by inequality?

If two real numbers or algebraic expressions are related using the symbols >, <, ≥, ≤, then the relation is called an inequality.

5. What is the importance of inequality math?

In mathematics, inequalities are used to compare the relative size of values. They can be used to compare integers, variables, and various other algebraic expressions.

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