#### Chapter 3 Inequalities in one Variable

Section 3.1 Inequalities and their Solution Sets

# 3.1.1 Introduction

##### Info 3.1.1

If two numbers are related by one of the comparators $\le$, $<$, $\ge$, or $>$, a statement is generated that can be true or false depending on the numbers:
• $a (reads: "$a$ is strictly less than $b$" or simply "$a$ is less than $b$") is true if the number $a$ is less than and not equal to $b$.
• $a\le b$ (reads: "$a$ is less than $b$") is true if $a$ is less than or equal to $b$.
• $a>b$ (reads: "$a$ is strictly greater than $b$" or simply "$a$ is greater than $b$") is true if the number $a$ is greater and not equal to $b$.
• $a\ge b$ (reads: "$a$ is greater than $b$") is true if the number $a$ is greater than or equal to $b$.

The comparators describe how the given values are related to each other on the number line: $a means that $a$ is to the left of $b$ on the number line.
##### Example 3.1.2
The statements $2<4$, $-12\le 2$, $4>1$, and $3\ge 3$ are true, but the statements $2<\sqrt{2}$ and $3>3$ are false.

On the number line, the number $2$ is to the left of the number $4$, thus $2<4$.

Here, $a means the same as $b>a$, likewise $a\le b$ means the same as $b\ge a$. But it should be noted that the opposite of the statement $a is the statement $a\ge b$ and not $a>b$. If terms with a variable occur in an inequality, the problem is to find the number range of the variable such that the inequality is true.