MathsTips.com

Maths Help, Free Tutorials And Useful Mathematics Resources

  • Home
  • Algebra
    • Matrices
  • Geometry
  • Trigonometry
  • Calculus
  • Business Maths
  • Arithmetic
  • Statistics
Home » Algebra » Linear Inequalities

Linear Inequalities

The mathematical statement which says that one quantity is not equal to another quantity is called Inequalities.

For example, if x and y are two quantities such that x \neq y then any one of the following four conditions will be true:

i.e., either i) x>y ii) x \geq y iii) x <y iv) x \leq y

Each of the four conditions, given above, is an inequality. In the same way each of the following also represents an inequality:

-x+4 \leq 3, x+8 >4, etc.

Linear inequality in one variable:

  1. ax+b >c
  2. ax+b <c
  3. ax+b \geq c
  4. ax+b \leq c

Solving Linear Inequality

To solve a given linear inequality means to find the value or values of the variable used in it.

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

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

e.g. 2x+3>7 \Rightarrow 2x> 7-3

Rule 2: On transferring a negative term from one side of an inequality to it’s other side, the sign of the term becomes positive.

e.g. 2x-3 >7 \Rightarrow 2x > 7+3

Rule 3: If each term of an inequality be multiplied or divided by the same positive number, the sign of the inequality remains the same.

e.g x< 6 \Rightarrow 4x < 4 \times 6

Rule 4: If each term of an inequality be multiplied or divided by the same negative number, the sign of inequality changes.

e.g x >5 \Rightarrow -3x <-3 \times 5

Rule 5: If signs of each term on both the sides of an inequality are changed, the sign of inequality reverses.

e.g -x >5 \Rightarrow x<-5

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

e.g x>y \Rightarrow \dfrac{1}{x} < \dfrac{1}{y}

Replacement set & Solution set

The set, from which the absolute value of the variable x is to be chosen, is called replacement set and its subset, whose elements satisfy the given inequality ,is called the solution set.

Example: Let the given inequality be x<3 :

  1. If the replacement set = \mathbb{N} , the set of natural numbers,the solution set = {1,2}
  2. If the replacement set = \mathbb{W} , the set of whole numbers, the solution set = {0,1,2}
  3. If the replacement set = \mathbb{Z} or \mathbb{I} ,the set of integers, the solution set ={ 2, 1, 0, -1, -2…}

Example 1: If the replacement set is the set of natural numbers (N) find the solution of: 3x+4< 16

Solution:

3x+4 <16 \\    or, 3x < 16-4 \\    or, 3x<12 \\    or, \dfrac{x}{3} < \dfrac{12}{3} \\    i.e., x<4

\because the replacement set = \mathbb{N} (natural numbers)

\therefore solution set= {1, 2, 3}

Example 2: If the replacement set is the set of whole numbers ( \mathbb{W} ), find the solution set of : 5x+4 \leq 24

Solution:

5x \leq 24-4 \\    or, 5x \leq 20 \\    or, x \leq \dfrac{20}{5} \\    or, x \leq 4

\because the replacement set is the set of whole numbers

\therefore the solution set = {0, 1, 2, 3, 4}

Example 3: If the replacement set is the set of integers between -6 and 8, find the solution set of 15-3x >x-3

Solution:

15-3x > x-3 \\    or, -3x-x > -3-15 \\    or, -4x > -18 \\    or, \dfrac{-4x}{-4} > \dfrac{-18}{-4} \\    or, x< 4.5

\because the replacement set is the set of integers between -6 and 8

\therefore the solution set = {-5,-4,-3,-2,-1, 0, 1, 2, 3, 4}

Example 4: If the replacement set is the set of real numbers (R), find the solution set of: 8+ 3x \geq 28-2x

Solution:

3x+ 2x \geq 28-8 \\    or, 5x \geq 20 \\    or, x \geq 4

\because the replacement set is the set of real numbers

\therefore solution set = { x: x \geq 4 and x \in \mathbb{R} }

Example 5: Solve the following inequality: 2y-3 < y+1 \leq 4y+7, if

  1. y \in \mathbb{Z} (integers)
  2. y \in \mathbb{R} (real numbers)

Solution:

2y-3>y+1 4 and y+1 \leq 4y+7

or, y <4 and -6 \leq 3y

or, y<4 and y \geq -2

or, -2 \leq y <4

When. y \in \mathbb{Z}

\therefore solution set = {-2, -1 ,0 ,1 ,2, 3}

When y \in \mathbb{R}

\therefore solution set = { y: -2 \leq y<4 and y \in \mathbb{R} }

Exercise

  1. State true or false:
    1. -5x \geq 15 \Rightarrow x \geq -3
    2. 7>5 \Rightarrow \dfrac{1}{7} < \dfrac{1}{5}
  2. State whether the following statements are true or false given a, b, c, d are real numbers and c \neq 0 .
    1. If a<b , then a-c< b-c
    2. If a>b , then a+c> b+c
    3. If a<b then ac> bc
    4. If a>b then \dfrac{a}{c} < \dfrac{b}{c}
    5. If a-c >b-d then a+d > b+c
  3. If x \in \mathbb{N} , find the solution set of the following inequalitys:
    1. 5x +3 \leq 2x+18
    2. 3x-2 < 19-4x
  4. If the replacement set is the set of whole numbers, solve:
    1. 7-3x \geq \dfrac{-1}{2}
    2. x-\dfrac{3}{2} < \dfrac{3}{2} -x
  5. Solve the inequality: 3-2x \geq x-12 given that x \in \mathbb{N} .
  6. If 25-4x <16 , find:
    1. The smallest value of x , when x is a real number.
    2. The smallest value of x , when x is an integer.
  7. If the replacement set is the set of real numbers, solve:
    1. 5+\dfrac{x}{4} > \dfrac{x}{5} +9
    2. \dfrac{x+3}{8} < \dfrac{x-3}{5}
  8. Find the smallest value of x for which 5-2x < 5\dfrac{1}{2} -\dfrac{5}{3}x , where x is an integer.
  9. Find the largest value of x for which 2(x-1) \leq 9-x and x \in \mathbb{W} .
  10. Solve the inequality: 12+ 1\dfrac{5}{6}x \leq 5+3x and x \in \mathbb{R} .
  11. Given x \in \mathbb{Z} (integers), find the solution set of: -5 \leq 2x-3 < x+2
  12. Given x \in \mathbb{W} (whole numbers), find the solution set of : -1 \leq 3+4x <23 .
« Integers
An Introduction to Fundamental Algebra »


Filed Under: Algebra Tagged With: Linear Equation, Linear Inequalities, Linear Inequation

Comments

  1. plory says

    January 26, 2020 at 7:26 pm

    hey why don’t i ask the questions and you reply you teach me what i know

    Reply
  2. plory says

    January 26, 2020 at 7:30 pm

    by the way it’s nice of you to let me overview my past but i have to go forward and not backward

    Reply

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Table of Content

  • Complex Numbers
  • Quadratic Equations
  • Logarithm
  • Permutation
  • Combination
  • More on Complex Numbers
  • Classification of Numbers
  • Positive and Negative Quantities
  • Understanding Simple Algebraic Formulas With Examples
  • Integers
  • Linear Inequalities
  • An Introduction to Fundamental Algebra
  • Basic Number Properties – Commutative, Associative and Distributive
  • Algebraic Multiplication and Division
  • Simple Equations in One Variable
  • Simple Formulae and their Application
  • Rational and Irrational Numbers
  • Problems Leading to Simple Equations
  • Simultaneous Equations
  • Mathematical Induction
  • Different Type of Sets
  • Indices
  • Framing Formulas
  • Sequences
  • Introduction to Matrices
  • Addition Of Matrices
  • Subtraction Of Matrix
  • Multiplication of Matrices
  • Determinant of Matrices
  • Co-factor of Matrices
  • Minor of Matrices
  • Transpose and Adjoint of Matrices
  • Inverse of a Matrix
  • System of Linear Equations in Matrices
  • Introduction to Polynomials
  • Classification of Polynomials
  • Addition and Subtraction of Polynomials
  • Multiplication of Polynomials
  • Factoring Polynomials
  • Zeroes of Polynomial
  • Remainder Theorem of Polynomials
  • Factor Theorem of Polynomial
  • Simplifying Polynomial Fractions
  • Roots of a Polynomial
  • Addition of Polynomial Fractions
  • Subtraction of Polynomial Fractions
  • Multiplying polynomial fractions
  • Division of Polynomial Fractions

© MathsTips.com 2013 - 2025. All Rights Reserved.