Properties of Inequalities: In mathematics, inequality occurs when two mathematical statements or two numbers are compared in a non-equal way. In general, inequalities can be either numerical or algebraic in nature or a combination of the two. When two linear algebraic expressions of degree \(1\) are compared, linear inequalities occur. Different types of linear inequalities can be represented in a variety of ways.

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. To solve inequalities, we must follow specific rules. In this article, we will learn about the properties of inequalities.

Properties of Linear Inequalities

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

1. The sign of a positive term becomes negative when it is transferred from one side of an inequation to the other. Example: \(3x+4>8⟹3x>8-4\)

2. On transferring a negative term from one side of an inequation to its other side, the sign of the term becomes positive. Example: \(21≥2x-13⟹21+13≥2x\)

3. If each term of an inequation is multiplied or divided by the same positive number, the inequality sign remains the same. That is if \(p\) is positive and \(p≠0.\)

(i) \(x < y \Rightarrow px < py\) and \(\left( {\frac{x}{p}} \right) < \left( {\frac{y}{p}} \right)\)
(ii) \(x > y \Rightarrow px > py\) and \(\left( {\frac{x}{p}} \right) > \left( {\frac{y}{p}} \right)\)
(iii) \(x \le y \Rightarrow px \le py\) and \(\left( {\frac{x}{p}} \right) \le \left( {\frac{y}{p}} \right)\)
(iv) \(x \ge y \Rightarrow px \ge py\) and \(\left( {\frac{x}{p}} \right) \ge \left( {\frac{y}{p}} \right)\)

Example: \(x \le 3 \Rightarrow 2x \le 3 \times 2,x \ge 7 \Rightarrow 4x \ge 7 \times 4,x \le 2 \Rightarrow \frac{x}{6} \le \frac{2}{6}\) and so on.

4. If each term of an inequation is multiplied or divided by the same negative number, the sign of inequality reverses. That is if \(p\) is negative.

(i) \(x < y \Rightarrow px > py\) and \(\left( {\frac{x}{p}} \right) > \left( {\frac{y}{p}} \right)\)
(ii) \(x \ge y \Rightarrow px \le py\) and \(\left( {\frac{x}{p}} \right) \le \left( {\frac{y}{p}} \right)\)

Example: \(x \le 4 \Rightarrow – 2x \ge – 2 \times 4,x > 3 \Rightarrow – 5x < – 5 \times 3, – 2x \ge 4 \Rightarrow \frac{{ – 2x}}{{ – 2}} \le \frac{4}{{ – 2}}\) and so on.

5. If the sign of each term on both sides of an inequation is changed, the sign of inequality gets reversed.

Example: \(3x \le 15 \Leftrightarrow – 3x \ge – 15\)

6. If both sides of an inequation are positive or both are negative, then the sign of inequality reverses on taking their reciprocals. 

i.e. if \(x\) and \(y\) both are either positive or both are negative, then

(i) \(x > y \Leftrightarrow \frac{1}{x} < \frac{1}{y}\)
(ii) \(x \le y \Leftrightarrow \frac{1}{x} \ge \frac{1}{y}\)
(iii) \(x \ge y \Leftrightarrow \frac{1}{x} \le \frac{1}{y}\) and so on.

Example: \(x \ge 3 \Leftrightarrow \frac{1}{x} \le \frac{1}{3}\)

Addition Property of Inequality

According to the addition property of linear inequality, adding the same number to each side of the inequality results in an equivalent inequality, i.e., the inequality symbol remains the same.

Example: \(x<2⟹x+3<2+3\) and \(y>x⟹y+b>x+b.\)

Subtraction Property of Inequality

According to the linear inequality subtraction property, removing the same number from either side generates an identical inequality, i.e., the inequality symbol remains the same.

Example: \(x<5⟹x-1<5-1\) and \(y>x⟹y-b>x-b.\)

Multiplication Property of Inequality

According to the linear inequality multiplication property, multiplying both sides of an inequality by a positive number always results in an equivalent inequality, i.e., the inequality symbol remains the same.

Example: \(x<2⟹x×3<2×3\)

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: \(x<2\)

Multiply the given inequation by \(-2.\)

Thus, \(-2x>-4\)

Division Property of Inequality

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: \(2x<4\)

Divide the given inequation by \(2.\)
Thus, \(\frac{{2x}}{2} < \frac{4}{2}\)
However, if the inequality sign is reversed, dividing both sides of an inequality with a negative number gives an equivalent inequality.

Example: \(-3x>9\)

Divide the given inequation by \(-3.\)
Thus, \(\frac{{-3x}}{-3} < \frac{9}{-3}.\)

Transitive Property of Inequality

The term “transitive” refers to the word transfer.

The transitive property states, “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.”

Example: If \(a<b\) and \(b<c\) then \(a<c.\)

Comparison Property of Inequality

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.

If \(a=b+c\) and \(c>0,\) then \(a>b\)

Example: \(8=5+3,\) then \(8>5.\)

Solved Examples – Properties of Inequalities

Q.1. Solve the following inequation for real \(x:\frac{{2x – 1}}{3} \ge \frac{{3x – 2}}{4} – \frac{{2 – x}}{5}\)
Ans: Given \(\frac{{2x – 1}}{3} \ge \frac{{3x – 2}}{4} – \frac{{2 – x}}{5}.\)
Multiplying both sides by \(60,\) L.C.M. of \(3,4\) and \(5,\) we get
\(20(2x – 1) \ge 15(3x – 2) – 12(2 – x)\)
\( \Rightarrow 40x – 20 \ge 45x – 30 – 24 + 12x\)
\( \Rightarrow 40x – 20 \ge 57x – 54\)
\( \Rightarrow 40x – 57x \ge – 54 + 20\)
\( \Rightarrow \, – 17x \ge – 34\)
(Dividing by \(-17)\)
\(⟹x≤2\)
As \(x∈R,\) the solution set\( = \{ x:x \in R,x \le 2\} \)

Q.2. \(P\) is the solution set of \(8x-1>5x+2\) and \(Q\) is the solution set of \(7x-2≥3x+6,\) where \(x∈N.\) Find the set \(P∩Q.\)
Ans: Given \(8x-1>5x+2\)
\( \Rightarrow 8x – 1 + ( – 5x + 1) > 5x + 2 + ( – 5x + 1)\,({\rm{Add}}\, – 5x + 1)\)
\(⟹3x>3\)
\(⟹x>1\) but \(x∈N,\)
Therefore, \(P = \{ 2,3,4,5, \ldots \} .\)
Also \( \Rightarrow 7x – 2 + ( – 3x + 2) \ge 3x + 18 + ( – 3x + 2)\,({\rm{Add}}\, – 3x + 2)\)
\(⟹4x≥20\)
\(⟹x≥5\) but \(x∈N,\)
Therefore, \(Q = \{ 5,6,7,8, \ldots \} .\)
Therefore, \(P \cap Q = \{ 5,6,7,8, \ldots \} .\)

Q.3. Show that the sign of inequality remains same if we add and subtract \(3\) and \(2\) respectively from the following inequalities
i. \(7<10\)
ii. \(5>-7\)
Ans: Given as \(7<10\)
\(⟹7+3<10+3\) (adding \(3\) on both sides)
\(⟹10<13\)
\(⟹10-2<13-2\) (subtracting \(2\) from both sides)
\(⟹8<11,\) which is true.
Hence, the sign of the given inequality remains the same.
Given as \(-5>-7\)
\(⟹-5+3>-7+3\) (adding \(3\) on both sides)<\br>
\(⟹-2>-4\)
\(⟹-2-2>-4-2\) (subtracting \(2\) from both sides)
\(⟹-4>-6,\) which is true.
Hence, the sign of the given inequality remains the same.

Q.4. Show that the sign of inequality remains the same if we multiply and divide the inequality \(15>6\) by \(2\) and \(6,\) respectively.
Ans: Given inequality is \(15>6\)
\(⟹15×2>6×2\)
[multiplying both sides by \(2]\)
\(⟹30>12\)
\( \Rightarrow \frac{{30}}{6} > \frac{{12}}{6}\)
[dividing both sides by \(6]\)
\(⟹5>2,\) which is true.
Hence, this shows that the sign of inequality remains the same if we multiply or divide into both sides of the inequality by the same positive number.

Q.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.
Ans: Let the integer be \(x,\) then one-third of the following integer is \(\frac{{x + 1}}{3}\) and one-fourth of the previous integer is \(\frac{{x – 1}}{4}.\)
According to the question,
\(\frac{{x + 1}}{3} \ge \frac{{x – 1}}{4} + 2\)
\( \Rightarrow \frac{{12(x + 1)}}{3} \ge \frac{{12(x – 1)}}{4} + 2 \times 12\) (multiplying both sides by \(12\))
\( \Rightarrow 4(x + 1) \ge 3(x – 1) + 24\)
\(⟹4x+4≥3x-3+24\)
\( \Rightarrow 4x + 4 – (3x + 4) \ge 3x + 21 – (3x + 4)\) (subtracting \((3x+4)\) from both sides)
\( \Rightarrow 4x + 4 – 3x – 4 \ge 3x + 21 – 3x – 4\)
\(⟹x≥17\)
Hence, the smallest value of \(x\) is \(17.\)

Summary

In this article, we learnt about properties of linear inequalities, addition property of inequality definition, subtraction property of inequality definition, multiplication property of inequality definition, division property of inequality definition, transitive property of inequality, comparison property of inequality, solved examples on properties of inequalities, and FAQs on properties of inequalities.

The learning outcome of this article is when determining the optimum solution to a problem, a system of linear inequalities is frequently used.

How to Represent Linear Inequality on Number Line?

FAQs

Q.1. What are the properties of inequality?
Ans: The following are the properties of linear inequalities:
i. The sign of a positive term becomes negative when it is transferred from one side of an inequation to the other.
ii. On transferring a negative term from one side of an inequation to its other side, the sign of the term becomes positive.
iii. If each term of an inequation is multiplied or divided by the same positive number, the inequality sign remains the same.
iv. If each term of an inequation is multiplied or divided by the same negative number, the sign of inequality reverses.
v. If the sign of each term on both sides of an inequation is changed, the sign of inequality gets reversed.
vi. If both sides of an inequation are positive or both are negative, then the sign of inequality reverses on taking their reciprocals.

Q.2. What are the basic properties of the addition of inequality?
Ans: According to the addition property of linear inequality, adding the same number to each side of the inequality results in an equivalent inequality, i.e., the inequality symbol remains the same.
Example: \(x<7⟹x+1<7+1\) and \(x>z⟹x+a>z+a.\)

Q.3. What is the distributive property of inequality?
Ans: Parentheses have a role in the inequalities you’ll see in this concept. The distributive property can be used to simplify an equation containing parentheses. You can do the same thing with inequalities. You can use the distributive property to simplify an inequality and make it easier to solve.
Example: Solve for \(x: -9(x+3)<45\)
To begin, apply the distributive property to the inequality’s left side. Add the products of multiplying each of the two integers inside the parentheses by \(-9.\)
That is, \(-9x-27<45\)
\(⟹-9x+(-27)+(27)<45-(27)\)
\(⟹-9x<18\)
\( \Rightarrow \frac{{ – 9x}}{{ – 9}} < \frac{{18}}{{ – 9}}\)
\(⟹x>-2\)
Therefore, \(x>-2.\)

Q.4. What are the 4 types of inequalities?
Ans: There are \(4\) types of inequalities:

Symbol	Meaning
\(<\)	Less than
\(>\)	Greater than
\(≤\)	Less than or equal
\(≥\)	Greater than or equal

Q.5. What are the basic properties of the division of inequalities?
Ans: 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: \(5x<15\)
Divide the given inequation by \(5.\)
Thus, \(\frac{{5x}}{5} < \frac{{15}}{5}.\)
If the inequality sign is reversed, however, dividing both sides of an inequality with a negative number gives an equivalent inequality.
Example: \(-2x>-8\)
Divide the given inequation by \(-2.\)
Thus, \(\frac{{ – 2x}}{{ – 2}} < \frac{{ – 8}}{{ – 2}} \Rightarrow x < 4.\)

We hope this detailed article on the properties of inequalities helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!