Linear Inequalities – GCSE Maths

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Introduction

  • Inequalities are similar to equations. Equations do have specific values that satisfies the equation on the hand Inequalities have feasible region( set of the values satisfying the inequality ) .
  • Example of Inequality-

Introduction image of linear inequalities with an equation explaining every element, constant, variable and symbols

  • Using Inequalities, we can solve real-life problems where the maximum or minimum quantity of something is calculated under multiple constraints.
  • They are basically used to represent those problems which are restricted by some constraint or conditions. That is why they are used in various fields like Business, Engineering and Economics etc.

Inequality is an expression in which variables and constants are present and one expression is lesser than the other.Image of list of symbols and their meanings

  • The Solution to these Inequalities exist in form of intervals (set of integers that lies between two numbers) –

Example: (-12,1] The number is greater than -12 and lesser than 1. Number line showing exclusion and inclusion for linear inequalities

Linear Inequalities

  • The Inequalities in which the maximum power raised to a variable is one.

Example:

2x + 9 < 15

Steps to Solve Linear Inequality: 

  • Step#1: Find value of the variable using addition, Subtraction, Multiplication or division on both sides of inequality so that the variable will get isolated.image of step 1 of instructions of step by step instructions of solving linear inequalities

image of step 2 of instructions of step by step instructions of solving linear inequalities Means that, x < 3

(Whenever we change sign on both the sides the symbol is reversed from greater than to lesser than and vice versa)

  • Step#2: Express the solution in form of interval or using number line.

x = (-∞,3)

image of Number line2 of instructions of step by step instructions of solving linear inequalities

certified Physics and Maths tutorSolved Example

Problem: Solve the following linear inequalities –

(1) 3y – 8 > 18            (2) 5 + 6y < 17

Solution:

(1) 3y – 8 > 18   

Step#1: 

  • Adding 8 on both sides-image of step 1 for linear inequalities solved example
  • Dividing both sides by 3-image of step 2 for linear inequalities solved example

Step#2: Representing the solution in form of interval- y = (6, ∞)

image of Number line for step by step solved example

(2) 5 + 6y < 17

Step#1: 

  • Subtracting 5 on both sides-image of step 1 for linear inequalities solved example
  • Dividing both sides by 6-image of step 2 for linear inequalities solved example

Step#2: Representing the solution in form of interval- y = (-∞, 2) image of Number line for step by step solved example

certified Physics and Maths tutorSolved Example:

Problem: Write the Inequalities that these number lines represents:

Solution: 

  1. Number line for step by step solved example for linear inequalities

Answer : (-2,2), Inequality = -2 < x < 2

2.

Number line for step by step solved example

Answer : (-4,1), Inequality = -4 ≤ x ≤ 1

3.

Number line for step by step solved example

Answer : (6,∞), Inequality = x > 6

4.

Number line for step by step solved example forinequalities

Answer : (3,7), Inequality = 3 ≤ x < 7

5.

Number line for step by step solved example

Answer : (6,10), Inequality = 6 ≤ x ≤ 7

 

certified Physics and Maths tutorSolved Example:

Problem: Solve the Inequality and represent the answer in interval form as well as on the number line.image of question for solved exampleSolution: 

Step#1:

  • Subtract 125 on both the sides-image of step 1 for linear inequalities solved example
  • Divide by -5 on both the sides-image of step 2 for linear inequalities solved example

(The symbols were changed that is why Inequality symbol is reversed from ‘>’ to ‘<’)

  • Interval : (-∞,5)
  • Number Line :image of Number line for step by step solved example

certified Physics and Maths tutorSolved Example:

Problem: Draw the number lines to show these Inequalities –

(1) x > 12

(2) 5 < x ≤ 7

(3) 3 < x < 5

(4) 10 ≤ x ≤ 13

(5) 15 ≤ x

Solution:

  1. x > 12

Number line for step by step solved example for linear inequalities

2. 5 < x ≤ 7Number line for step by step solved example

3. 3 < x < 5
Number line for step by step solved example for linear inequalities
4. 10 ≤ x ≤ 13
Number line for step by step solved example
5. 15 ≤ x
Number line for step by step solved example for linear inequalities
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