Lesson 6: Understand and Write Inequalities

Property

An inequality is a statement that compares two expressions, asking for which values of the unknowns the comparison is true.
The set of all such values is the solution set.
The main types of inequalities are:

• $A < B$ ($A$ is less than $B$)
• $A > B$ ($A$ is greater than $B$)
• $A \leq B$ ($A$ is at most $B$, or less than or equal to $B$)
• $A \geq B$ ($A$ is at least $B$, or greater than or equal to $B$)

Examples

• Does $x=10$ make the inequality $x > 8$ true? Yes, because 10 is greater than 8.
• Does $x=7$ make the inequality $x \leq 7$ true? Yes, because 7 is equal to 7, which fits the 'less than or equal to' condition.
• Does $x=3$ make the inequality $x < 3$ true? No, because 3 is not strictly less than 3.

Property

To write an inequality from a verbal phrase, you match keywords to their corresponding inequality symbols.

• Less than ($<$): "is less than", "is smaller than"
• Greater than ($>$): "is greater than", "is more than"
• Less than or equal to ($\leq$): "is at most", "is no more than", "maximum"
• Greater than or equal to ($\geq$): "is at least", "is no less than", "minimum"

Property

The "not equal to" symbol, $\neq$, is used to show that a variable cannot be a specific value. An inequality like $x \neq c$ means that $x$ can be any number except for $c$.

Examples

• The number of students, $s$, is not 25: $s \neq 25$
• The temperature, $t$, is anything but $0^\circ$: $t \neq 0$
• The value of $y$ is not equal to -10: $y \neq -10$

Explanation

The "not equal to" symbol, $\neq$, is used when a specific value is excluded from the set of possibilities. Unlike other inequality symbols that define a range (like greater than or less than), this symbol only states what the variable cannot be. For example, if the number of players on a team, $p$, cannot be 11, we write $p \neq 11$. This means the team could have any number of players, just not exactly 11.