7th Grade - Linear Inequalities in One Variable

Introduction

  • An inequality is a mathematical sentence that makes a non-equal comparison between two mathematical expressions.
  • There are four inequality operators, as shown in the table below:
Operator Symbol
Less than <
Graeter than >
Less than or equal to
Greater than or equal to
  • Examples of inequality include: 2x>45y<204m-26, etc.

Solving simple linear inequalities

  • Inequalities can be solved using the rules mentioned below.
  • Addition Rule:
    • Adding the same number to each side of an inequality produces an equivalent inequality wherein the sign of inequality remains unaffected.
  • Subtraction Rule:
    • Subtracting the same number from both sides of an inequality results in an equivalent inequality wherein the sign of inequality remains unaffected.
  • Multiplication Rule:
    • Multiplying both sides of an inequality by a positive number results in an equivalent inequality wherein the sign of inequality remains unaffected.
    • Multiplying both sides of an inequality by a negative number results in a new inequality wherein the sign of inequality is reversed.
  • Division Rule: 
    • Dividing both sides of an inequality by a positive number results in an equivalent inequality wherein the sign of inequality remains unaffected.
    • Dividing both sides of an inequality by a negative number results in a new inequality wherein the sign of inequality is reversed.

Graphical Representation of Linear Inequalities in One Variable

  • Linear inequalities in one variable can be visualized on a number line.
  • Plotting linear inequalities in one variable involves the following steps:
    • First, solve the linear inequality to find the variable's value that satisfies the inequality.
    • Locate and mark the solution on the number line.
  • We use an open dot for strict inequalities to indicate that the point is not part of the solution.
  • We use a closed dot for slack inequalities to indicate that the point is part of the solution.

Solved Examples

Question 1: Solve the following inequality.

m-25

Solution: m-25

Applying addition rule

m-2+25+2

m7

Question 2: Solve the following inequality.

x+3>1

Solution: x+3>1

Applying subtraction rule

x+3-3>1-3

x>-2

Question 3: Solve the following inequality.

x24

Solution: x24

Apply multiplication rule (by a positive number)

x2×24×2

x8

Question 4: Solve the following inequality.

-m43

Solution: -m43

Apply multiplication (by a negative number) rule

-m4×-43×-4 [Note that the inequality sign reversed due to multiplication by a negative number]

m-12

Question 5: Solve the following equation.

7x<21

Solution: 7x<21

Apply the division rule (by a positive number)

7x7<217

x<3

Question 6: Solve the following inequality.

2x8

Solution: 2x8

Apply division (by a negative number) rule

2x282

x4

Question 7: What inequality does the number line shown below represents?

Solution: An open circle is used to signify "less than (<)" or "greater than (>)" type inequality on the number line. The given figure represents x<10.

Question 8: What inequality does the number line shown below represents?

Solution: A closed circle denotes "less than equal to ()" or "more than equal to ()" type inequality on the number line. The given figure represents x-1.

Cheat Sheet

  • If two mathematical expressions are separated by symbols such as >, <, , it represents an inequality.
  • To solve an inequality, we can apply the rules of addition, subtraction, multiplication, and division.
  • Adding the same quantity to both sides of an inequality doesn't change the direction of the inequality sign.
  • Likewise, subtracting the same quantity from both sides of an inequality doesn't change the direction of the inequality sign.
  • Multiplying both sides of an inequality by a positive number leaves the inequality sign unchanged. In contrast, multiplication by a negative number reverses the direction of inequality.
  • Similarly, dividing both sides of an inequality by a positive number leaves the inequality sign unchanged. At the same time, division by a negative number reverses the direction of the inequality.
  • Linear inequality in one variable is plotted on a number line. An open circle is used to signify "less than (<)" or "greater than (>)", while a closed circle denotes "less than equal to ()" or "more than equal to ()" type inequality on the number line.

Blunder Areas

  • When solving inequalities, one must not forget to reverse the inequality sign when multiplying or dividing both sides by negative numbers.