6.06 Write inequality statements
Introduction
So far we have learned about equations, where both sides have the same value. It is also possible to compare two sides of an expression where one side is smaller or larger than the other.
We've previously used inequalities to compare numbers . We can also use inequalities to compare two sides of an expression when one side of the expression is greater than the other. We can think of these expressions as a set of unbalanced scales, where one side is heavier than the other.
Ideas
Inequality symbols
The greater than symbol \gt indicates that the expression on the left of the symbol has a larger value than the expression on the right of the symbol. For example, 3\gt2 means "3 is greater than 2".
Similarly, the less than symbol \lt indicates that the expression on the left of the symbol has a smaller value than the expression on the right of the symbol. For example, 3\lt4 means "3 is less than 4".
The greater than or equal to symbol \geq indicates that the expression on the left of the symbol has a larger value than or is equal to the expression on the right of the symbol. So we could write 3\geq2, as well as 2\geq2, or n\geq2. This last expression "x is greater than or equal to 2" is true for a whole range of values of n, such as n=2, n=2.5 and n=11.
Similarly, the less than or equal to symbol \leq indicates that the expression on the left of the symbol has a smaller value than or is equal to the expression on the right of the symbol. So we could write 3\leq4, as well as 4\leq 4, or n\leq4. Once again, the last expression "x is less than or equal to 4" is true for a whole range of values of x, such as x=4, x=3.5 and x=-1.
The smaller side of the inequality symbol matches the side with the smaller number. That is, the inequality symbol "points to" the smaller number.
The images below show another demonstration of the inequality symbols:
We are familiar with being able to write an equation in two orders. For example, x=10 and 10=x mean the same thing.
We can also write inequality statements in two orders, but we now need to be careful and switch the inequality sign being used as well.
For example, x\gt10 means the same thing as 10\lt{x}. That is, "x is greater than ten" is the same as "ten is less than x".
Examples
Example 1
Write the following sentence using mathematical symbols: n is greater than 9.
Example 2
Choose the mathematical symbol that makes this number sentence true: \dfrac{2}{3} \enspace ⬚ \enspace 0.3
Example 3
If x\gt15, what is the smallest integer value x can have?
Inequalities are mathematical sentences where two expressions are not necessarily equal, indicated by the symbols: <, >, \leq, and \geq.
| Symbol | Meaning | Example |
|---|---|---|
| < | \text{less than} | 3<6 |
| > | \text{greater than} | 6>3 |
| \leq | \text{less than or equal to} | 4\leq 6 |
| \geq | \text{greater than or equal to} | 6\geq 5 |
The smaller side of the inequality symbol matches the side with the smaller number. That is, the inequality symbol "points to" the smaller number.