Graphing Solutions
To graph solutions, there are two parts to consider. First, the idea discussed above about the number being allowed or not allowed as a solution. An open circle is used to signify when a number is not a solution, and a closed circle is used to signify when a number is a solution:
| Symbol | Meaning | Graphing |
|---|---|---|
| \(<\) | "less than" | Use open circle \(\circ\) |
| \(>\) | "greater than" | Use open circle \(\circ\) |
| \(\le\) | "less than or equal to" | Use closed circle \(\bullet\) |
| \(\ge\) | "greater than or equal to" | Use closed circle \(\bullet\) |
Second, on a number line indicate which numbers are solutions with an arrow. For numbers smaller, the arrow will point to the left and for numbers larger, the arrow will point to the right. The following table shows some examples:
| Inequality | Arrow Direction & Circle | Graphed Solution |
|---|---|---|
| \(x\le5\) | points left closed circle |
|
| \(x> -3\) | points right open circle |
|
| \(-1< x\) | points right open circle |
|
VERY IMPORTANT NOTE:
The last example in the table above shows a less than problem, but the arrow graphs to the right?? This is due to the fact that the variable \(x\) is listed on the right side of the inequalitiy instead of on the left. A very common error is to assume less than problems always have arrows left and greater than problems always have arrows right. This is only true if the variable is on the left side of the inequality. To help avoid this confusion, always read the inequality from left to right. In the example in the table, \(-1< x\) would read as "\(-1\) is less than any number \(x\)", making \(-1\) the smallest number allowed, so all number larger are the solution.
