Graphing inequalities on a number line is a visual way to represent all the values that satisfy the inequality. Here's a breakdown of the process:
1. Understand the Inequality Symbol:
Before you start, make sure you understand what each inequality symbol means:
<: Less than (not including the endpoint)>: Greater than (not including the endpoint)≤: Less than or equal to (including the endpoint)≥: Greater than or equal to (including the endpoint)
2. Locate the Critical Value:
The critical value is the number in the inequality. For example, in the inequality x > 3, the critical value is 3. Find this number on the number line.
3. Determine the Type of Circle:
- Open Circle (o): Use an open circle when the inequality does not include the endpoint (i.e.,
<or>). This indicates that the critical value itself is not a solution to the inequality. - Closed Circle (●): Use a closed circle when the inequality does include the endpoint (i.e.,
≤or≥). This indicates that the critical value is a solution to the inequality.
4. Shade the Number Line:
- Greater Than (>) or Greater Than or Equal To (≥): Shade the number line to the right of the critical value. This represents all numbers greater than the critical value.
- Less Than (<) or Less Than or Equal To (≤): Shade the number line to the left of the critical value. This represents all numbers less than the critical value.
Example 1: Graphing x > 2
- Critical value: 2
- Symbol:
>(greater than, not including the endpoint) - Type of circle: Open circle at 2
- Shade: To the right of 2
The graph will show an open circle at 2, with the number line shaded to the right.
Example 2: Graphing x ≤ -1
- Critical value: -1
- Symbol:
≤(less than or equal to, including the endpoint) - Type of circle: Closed circle at -1
- Shade: To the left of -1
The graph will show a closed circle at -1, with the number line shaded to the left.
Summary Table:
| Inequality Symbol | Circle Type | Shading Direction | Endpoint Included? |
|---|---|---|---|
< |
Open | Left | No |
> |
Open | Right | No |
≤ |
Closed | Left | Yes |
≥ |
Closed | Right | Yes |
In short, to graph an inequality on a number line, identify the critical value, use an open or closed circle based on the inequality symbol, and shade the number line in the correct direction to represent the solution set.
