Introduction

An inequality is a mathematical statement that represents a set of numbers on the number line. For example, the set of numbers that are greater than or equal to 0 can also be represented by the inequality

x \; {\color{red}\geq} \; 0

Here, the sign {\color{red}\geq} means "greater than or equal to." So the above statement in words is " x is greater than or equal to zero." For example,

  • the numbers 0, 1, and 7.321 all satisfy the inequality, while
  • the numbers -0.01, -8, and -123 do not.

We can represent this inequality visually using a number line. To do so, we first plot the number zero as a red dot on the number line. The rest of the numbers that are greater than or equal to 0 lie to the right of 0, so we represent them using a red line that extends to the right of 0\mathbin{:}



In general, there are four types of inequalities:

\ge\quad greater than or equal - this is represented by a filled circle \Large\color{red}\bullet and an arrow that goes to the right \Large\color{red}\boldsymbol{\rightarrow}

>\quad greater than - this is represented by an empty circle \Large\color{red}\circ and an arrow that goes to the right \Large\color{red}\boldsymbol{\rightarrow}

\le\quad less than or equal - this is represented by a filled circle \Large\color{red}\bullet and an arrow that goes to the left \Large\color{red}\boldsymbol{\leftarrow}

<\quad less than - this is represented by an empty circle \Large\color{red}\circ and an arrow that goes to the left \Large\color{red}\boldsymbol{\leftarrow}

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Example: Finding the Inequality Represented by a Number Line

What inequality corresponds to the red colored graph shown in the number line below?


EXPLANATION

The red arrow indicates that x should consist of all the values greater than 2. In addition, the filled red circle at 2 implies that x can be equal to 2.

Therefore, the highlighted region represents the numbers that are greater than or equal to 2. This corresponds the inequality x \geq 2.

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Practice: Finding the Inequality Represented by a Number Line

Which inequality corresponds to the highlighted region shown on the number line above?

a
 $x \leq 1$
b
 $x \lt 1$
c
 $x \geq 1$
d
 $x > 1$
e
 $x \leq 0$
Practice: Finding the Inequality Represented by a Number Line

Which inequality corresponds to the highlighted region shown on the number line above?

a
 $x < -1$
b
 $x \geq -1$
c
 $x \leq -1$
d
 $x \leq 1$
e
 $x > -1$
Example: Constructing the Number Line for a Given Inequality

Represent x < -1 on a number line.

EXPLANATION

The symbol " < " means "less than." So, the inequality x < -1 represents the numbers that are less than -1.

On a number line, the numbers less than -1 are the numbers that are left of -1. So, we should draw an arrow going to the left of -1.

However, the number -1 itself is not less than -1, so we should not include it in our representation. To exclude -1, we draw an empty circle around -1.



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Practice: Constructing the Number Line for a Given Inequality

Which of the following number lines represents the inequality $x < 3?$

a
b
c
d
e
Practice: Constructing the Number Line for a Given Inequality

Which of the following number lines represents the inequality $x \geq -5?$

a
b
c
d
e
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