Inequality

Background

We talked about finding particular values on a number line, but how do we express a range of values on it?

Question

Graph:

1. x < 1
2. 
3. x > -2
4. 
5.  
6.  

Answer

See below for the graphs:

Analysis

When we're looking at a range of values as a solution to an expression, we indicate that a number is a solution by putting a solid dot over that particular value. For instance, if x = 0, we can put a dot over the number 0 on the number line:


If we have a series of numbers right next to each other, and the series of the numbers are real numbers (so there is no gap on the number line), we can indicate a range by having a series of dots one next to the other - and with them being right next to each other, we end up with a line.

For instance, for Question 1, we want to graph all the values on the number line that is less than 1 (we have the "less than" sign there). We'll therefore show a ray that starts at 1 and heads off towards smaller and smaller numbers. Since we don't want 1 to be a part of the solution, we don't put the full dot over it. What we do, however, is put a hollow dot over the 1 to show that the ray starts there but isn't part of the solution:



For Question 2, we do want the 1 to be part of the solution (the symbol means "less than or equal to", so we put a filled-in dot over the 1:



For Question 3, we do the same type of thing we did in question 1, but now we have the "greater than" sign. We put a hollow dot over -2 and run a ray towards bigger numbers:





And for Question 4, we put a filled dot over the -2:



Question 5 combines the two ranges we've been working with. It reads  and means we want all the values of x that are above or equal to -2 but less than 1. We can express it on the number line this way:



Question 6 reverses the direction of the signs. So let's look at it a little closer:




This says we want to graph all the points (x) that are either less than or equal to -2 or are greater than 1. That looks like this:



Oftentimes, because this notation is a bit confusing (with the two lines heading off in opposite directions), we'll write it as two separate statements. In this case we could say:



Vocabulary used:

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Numbers:

• Variables

Relations:

• Greater/Lesser Than
• Equality

Graphing:

• Number Line

Where might we go?