Linear Inequalities: In mathematics, inequality occurs when a non-equal comparison is made between two mathematical expressions or two numbers. In general, inequalities can be either numerical inequality or algebraic inequality. Numerical inequality occurs when two numbers are compared on the number line depending on their value. Algebraic inequality occurs when one expression is greater than or lesser than another expression. There are several ways to represent various kinds of inequalities.

Between any two quantities (say \(x\) and \(y\)) in the universe, only three conditions hold which are \(‘ < ‘,\,’ > ‘\) and \(‘ = ‘,\) i.e. \(x < y,\,x > y\) and \(x = y.\) The symbols \(‘ > ‘,\,’ < ‘,\,’\, \le ‘,\,’ \ge ‘\) are called the sign of inequality or inequation. In this article, we will learn everything about linear equalities.

Linear Inequalities Definition

Linear inequalities are defined as expressions in which two values are compared using the inequality symbols. An inequation is said to be linear if the exponent of each variable occurring in it is first degree only, and there is no term involving the product of the variables.

We can define inequations as a statement involving variable(s) and the sign of inequality \( > ,\, < ,\,\, \le ,\, \ge \) or two real numbers or two algebraic expressions related by the symbols \( > ,\, < ,\,\, \le ,\, \ge \) form an inequality.

If \(x\) and \(y\) are two quantities, then both of these quantities will satisfy any one of the following four conditions (relations): i.e., either \(x < y,\,x > y,\,x \le y\) or \(x \ge y.\) Each of the four conditions given above is an inequation.

Linear Inequality in One Variable

If \(a, b\) and \(c\) are real numbers, then each of the following is called a linear inequation in one variable:

1. \(ax + b > c.\) Read as : \(ax + b\) is greater than \(c.\)
2. \(ax + b < c.\) Read as : \(ax + b\) is less than \(c.\)
3. \(ax + b \ge c.\) Read as : \(ax + b\) is greater than or equal to \(c.\)
4. \(ax + b \le c.\) Read as : \(ax + b\) is less than or equal to \(c.\)

In an inequation, the signs \(‘ > ‘,\,’ < ‘,\,’ \ge ‘\) and \(‘ \le ‘\) are called signs of inequality.

Linear Inequations Example

A linear inequation with only one variable is called linear inequation in one variable and can be written as \(ax + b < 0\) or \(ax + b > 0\) or \(ax + b \ge 0\) or \(ax + b \le 0\) where \(a,b\) are real number and \(a \ne 0.\)
\(ax \le 0,\,ax + by + c > 0,\,ax \ge 5\) by \(x \le 10,\) etc are few examples of linear inequations.

Linear Inequalities Rules

Solving a Linear Inequation Algebraically: To solve a given linear inequation means to find the value or values of the variable used in it. Thus, to solve the inequation \(3x + 5 > 8\) means to find the variable \(x\) and to solve the inequation \(8 – 5y < 3\) means to find the variable \(y\) and so on.

The following working rules must be adopted for solving a given linear inequation:

Rule 1: On transferring a positive term from one side of an inequation to its other side, the sign of the term becomes negative.

Rule 2: On transferring a negative term from one side of an inequation to its other side, the sign of the term becomes positive.

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

1. \(x < y \Rightarrow px < py\) and \(\left({\frac{x}{p}} \right) < \left({\frac{y}{p}} \right)\)
2. \(x > y \Rightarrow px > py\) and \(\left({\frac{x}{p}} \right) > \left({\frac{y}{p}} \right)\)
3. \(x \le y \Rightarrow px \le py\) and \(\left( {\frac{x}{p}} \right) \le \left( {\frac{y}{p}} \right)\)
4. \(x \ge y \Rightarrow px \ge py\) and \(\left( {\frac{x}{p}} \right) \ge \left( {\frac{y}{p}} \right)\)

Rule 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.

1. \(x < y \Rightarrow px > py\) and \(\left({\frac{x}{p}}\right) > \left({\frac{y}{p}} \right)\)
2. \(x \ge y \Rightarrow px \le py\) and \(\left( {\frac{x}{p}} \right) \le \left( {\frac{y}{p}} \right)\)

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

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

That is if \(x\) and \(y\) both are either positive or both are negative, then:

1. \(x > y \Leftrightarrow \frac{1}{x} < \frac{1}{y}\)
2. \(x \le y \Leftrightarrow \frac{1}{x} \ge \frac{1}{y}\)
3. \(x \ge y \Leftrightarrow \frac{1}{x} \le \frac{1}{y}\) and so on.

Replacement Set and Solution Set

Replacement Set: The set from which values of the variable involved in the inequality are chosen is called the replacement set. Generally, we use either \(N\) (set of natural numbers; \(1,2,3,4,….\) ) or \(W\) (set of whole numbers; \(0,1,2,3,….\)) or \(I\) (set of integers; \(… – 3, – 2, – 1,0,1,2,3,….\)) or \(R\) (real number) as the replacement set.

Solution Set: The solution to an inequation is a number that, when substituted for the variable, makes the inequation true. The set of all solutions of the given inequation is called the solution set of the inequation. The solution set is the subset of the replacement set.

Method to Solve a Linear Inequation in One Variable

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 \(1.\)
4. Choose the solution set from the replacement set.

Representation of Solution Set of Linear Inequation in One Variable on Number Line

Use the following rules to represent the solution of a linear inequation in one variable on the number line 

1. If the inequation involves \( \le \) or \( \ge \) drawing a filled circle or dark circle \(\left( . \right)\) on the number line, then the number corresponding to the filled circle or dark circle is included in the solution set.
For example: If \(2\) is also included, i.e., \(x \le 2,\) then the circle will be darkened, and the graph will be as shown below:

2. If the inequation involves \( > \) or \( < ,\) then draw an unfilled circle or blank circle \(\left( o \right)\) on the number, the line shows that the number corresponding to the unfilled circle or blank circle is excluded from the solution set.
For example \(x < 2\) and \(x \in R\)

Solved Examples – Linear Inequalities (Practice Problems)

Q.5. If the replacement set is the set of integers, (\(I\) or \(Z\)), between \( – 6\) and \(8,\) find the solution set of \(15 – 3x > x – 3.\)
Ans: The given inequation is \(15 – 3x > x – 3.\)
\( \Rightarrow \, – 3x – x > – 3 – 15\)
\( \Rightarrow \, – 4x > – 18\)
\( \Rightarrow \frac{{ – 4x}}{{ – 4}} < \frac{{ – 18}}{{ – 4}}\) (Division by a negative number reverse the sign of inequality).
\( \Rightarrow x < 4.5\)
Since the replacement set is the set of integers between \( – 6\) and \(8.\)
Therefore, solution set \( = \left\{{ – 5, – 4, – 3, – 2, – 1,0,1,2,3,4} \right\}.\)

Q.1. If the replacement set is the set of natural numbers \(\left( N \right),\) find the solution set of \(3x + 4 < 16.\)
Ans: The given inequation is \(3x + 4 < 16\)
\( \Rightarrow 3x < 16 – 4\)
\( \Rightarrow 3x < 12\)
\( \Rightarrow \frac{{3x}}{3} < \frac{{12}}{3}\)
i.e. \(x < 4\)
Since the replacement set \( = N\) (Set of natural numbers)
Therefore, solution set \( = \left\{{1,2,3} \right\}.\)

Q.2. \(\frac{x}{2} – 5 \le \frac{x}{3} – 4,\) where \(x\) is a positive odd integer.
Ans: The given inequation is \(\frac{x}{2} – 5 \le \frac{x}{3} – 4,\)
\( \Rightarrow \frac{x}{2} – \frac{x}{3} \le – 4 + 5\)
\( \Rightarrow \frac{{3x – 2x}}{6} \le 1\)
\( \Rightarrow x \le 6\)
Since \(x\) is a positive odd integer
Therefore, solution set \( = \left\{{1,3,5} \right\}.\)

Q.3. Given that \(x \in R,\) solve the following inequality and graph the solution on the number line: \( – 1 \le 3 + 4x < 23.\)
Ans: Given: \( – 1 \le 3 + 4x < 23;\,x \in R\)
\( \Rightarrow \, – 1 \le 3 + 4x\) and \(3 + 4x < 23\)
\( \Rightarrow \, – 4 \le 4x\) and \(4x < 20\)
\( \Rightarrow \, – 1 \le x\) and \(x < 5\)
\( \Rightarrow \, – 1 \le x < 5;\,x \in R\)
Therefore, solution \(\left\{ { – 1 \le x < 5;\,x \in R} \right\}\)
The solution on the number line is

Q.4. List the solution set of \(50 – 3\left({2x – 5} \right) < 25,\) given that \(x \in W.\) Also, represent the solution set obtained on a number line.
Ans: Given: \(50 – 3\left({2x – 5} \right) < 25\)
\( \Rightarrow 50 – 6x + 15 < 25\)
\( \Rightarrow 65 – 6x < 25\)
\( \Rightarrow \, – 6x < 25 – 65\)
\( \Rightarrow \, – 6x < – 40\)
\( \Rightarrow \frac{{ – 6x}}{{ – 6}} > \frac{{ – 40}}{{ – 6}}\)(Division by a negative number reverse the sign of inequality).
\( \Rightarrow x > 6\frac{2}{3}\)
Required solution set \( = \left\{{7,8,9……} \right\}\)
And, the required number line is

Summary

In this article, we learnt about the definition of linear inequalities, linear inequations example, linear inequalities rules, replacement set and solution set, the method to solve a linear inequation in one variable, representation of solution set of linear inequation in one variable on the number line etc. There were also solved examples on linear inequalities and frequently asked questions on the same.

This article’s learning outcomes are that the system of linear inequalities is used to determine the best solution to a problem.

Frequently Asked Questions (FAQ) – Linear Inequalities

Q.1. Explain linear inequalities with an example?
Ans: An inequation is said to be linear if and only if the exponent of each variable occurring in it is first degree only. There is no term involving the product of the variables.
If \(a,b\) and \(c\) are real numbers, then each of the following is called a linear inequation in one variable.
Example: \(2x + 1 \le 2,\,2x + 3y + 5 > 7x + 2,\,5x \ge 5,\,y \le 10.\)

Q.2. How do I solve an inequality problem?
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 \(1.\)
4. Choose the solution set from the replacement set.

Q.3. How to represent the solution set of linear inequation on the number line?
Ans: Use the following rules to represent the solution of a linear inequation in one variable on the number line
1. If the inequation involves \( \le \) or \( \ge \) drawing a filled circle or dark circle \(\left( . \right)\) on the number line, the number corresponding to the filled circle or dark circle is included in the solution set.
2. If the inequation involves \( > \) or \( < ,\) then draw an unfilled circle or blank circle \(\left( o \right)\) on the number, the line to show that the number corresponding to the unfilled circle or blank circle is excluded from the solution set.

Q.4. What is a replacement set?
Ans: Replacement Set: The set from which values of the variable involved in the inequality are chosen is called the replacement set. Generally, we use either \(N\) (set of natural numbers; \(1,2,3,4,…..\)) or \(W\) (set of whole numbers; \(0,1,2,3,….\)) or \(I\) (set of integers;\(….. – 3,\, – 2,\, – 1,\,0,\,1,\,2,\,3,\,….\)) or \(R\) (real number) as the replacement set.

Q.5. How to solve linear inequalities?
Ans The following working rules must be adopted for solving a given linear inequation:
Rule 1: On transferring a positive term from one side of an inequation to its other side, the sign of the term becomes negative.
Rule 2: On transferring a negative term from one side of an inequation to its other side, the sign of the term becomes positive.
Rule 3: If each term of an inequation is multiplied or divided by the same positive number, the sign of inequality remains the same.
Rule 4: If each term of an inequation is multiplied or divided by the same negative number, the sign of inequality reverses.
Rule 5: If the sign of each term on both sides of an inequation is changed, the sign of inequality gets reversed.
Rule 6: If both the sides of an inequation are positive or both are negative, then on taking their reciprocals, the sign of inequality reverses.

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