Key Concepts
Inequality Signs
The box below shows the symbol, meaning, and an example for each inequality sign.
Symbol | Words | Example |
---|---|---|
≠ | not equal to | 2≠8, 2 is not equal to 8 |
> | greater than | 5>1, 5 is greater than 1 |
< | less than | 2<11, 2 is less than 11 |
≥ | greater than or equal to | 4≥4, 4 is greater than or equal to 4 |
≤ | less than or equal to | 7≤9, 7 is less than or equal to 9 |
The table below describes all the possible inequalities that can occur and how to write them using interval notation, where a and b are real numbers.
Inequality | Words | Interval Notation |
---|---|---|
a<x<b | all real numbers between a and b, not including a and b | (a,b) |
x>a | All real numbers greater than a, but not including a | (a,∞) |
x<b | All real numbers less than b, but not including b | (−∞,b) |
x≥a | All real numbers greater than a, including a | [a,∞) |
x≤b | All real numbers less than b, including b | (−∞,b] |
a≤x<b | All real numbers between a and b, including a | [a,b) |
a<x≤b | All real numbers between a and b, including b | (a,b] |
a≤x≤b | All real numbers between a and b, including a and b | [a,b] |
x<a or x>b | All real numbers less than a or greater than b | (−∞,a)∪(b,∞) |
All real numbers | All real numbers | (−∞,∞) |
The following table illustrates how the multiplication property is applied to inequalities, and how multiplication by a negative reverses the inequality:
Start With | Multiply By | Final Inequality |
a>b | c | ac>bc |
5>3 | 3 | 15>9 |
a>b | −c | −ac<−bc |
5>3 | −3 | −15<−9 |
The following table illustrates how the division property is applied to inequalities, and how dividing by a negative reverses the inequality:
Start With | Divide By | Final Inequality |
a>b | c | ac>bc |
4>2 | 2 | 42>22 |
a>b | −c | −ac<−bc |
4>2 | −2 | −42<−22 |