Graph This Inequality on a Number Line
Graphing an inequality on a number line is a fundamental skill in algebra. It shows which values satisfy an inequality and, just as importantly, which values do not. Because of that, whether you are working with (x>5), (x\leq -2), or a compound inequality such as (-3<x\leq 4), the number line provides a visual representation of every possible solution. By learning how to identify the boundary point, choose the correct type of circle, and shade the appropriate region, you can graph any inequality accurately and interpret the result with confidence.
What Does It Mean to Graph an Inequality on a Number Line?
An inequality describes a relationship between two expressions that is not represented by an equals sign. Common inequality symbols include:
- (<) meaning “less than”
- (>) meaning “greater than”
- (\leq) meaning “less than or equal to”
- (\geq) meaning “greater than or equal to”
To graph an inequality on a number line, you must identify the boundary where the inequality changes from true to false. You then represent that boundary with either:
- A closed circle if the boundary value is included
- An open circle if the boundary value is excluded
After marking the boundary, shade the side of the number line containing all the values that satisfy the inequality.
The shaded section represents the solution set: every point included in the shading is a possible value for the variable.
Understanding the Four Inequality Symbols
The meaning of the inequality symbol determines both the type of circle and the direction of the shading.
| Inequality | Meaning | Number-Line Symbol |
|---|---|---|
| (x<a) | (x) is less than (a) | Open circle at (a), shade left |
| (x>a) | (x) is greater than (a) | Open circle at (a), shade right |
| (x\leq a) | (x) is less than or equal to (a) | Closed circle at (a), shade left |
| (x\geq a) | (x) is greater than or equal to (a) | Closed circle at (a), shade right |
A helpful way to remember the direction is to imagine that the open side of the inequality symbol points toward the values that satisfy it.
- In (x<5), the symbol points toward the smaller numbers, so shade to the left.
- In (x>5), the symbol points toward the larger numbers, so shade to the right.
Steps to Graph an Inequality on a Number Line
1. Identify the Variable and Inequality
Begin by determining which quantity is unknown. Most number-line inequalities use (x), but the variable could also be (y), (n), (t), or another letter.
For example:
[ x\geq 3 ]
The variable is (x), and the inequality says that (x) must be at least (3).
2. Find the Boundary Point
Locate the number after the inequality symbol. This is the boundary point, sometimes called the critical value.
For (x\geq 3), the boundary is (3). Draw the number line and mark this point accurately Not complicated — just consistent..
3. Choose a Closed or Open Circle
Use the inequality symbol to decide whether the boundary point is included:
- If the symbol is (\leq) or (\geq), draw a closed circle.
- If the symbol is (<) or (>), draw an open circle.
A closed circle means the boundary value is part of the solution. An open circle means the boundary value is not part of the solution.
For (x\geq 3), use a closed circle at (3) Easy to understand, harder to ignore..
4. Shade the Correct Direction
Shade the side of the number line that represents the satisfying values.
Because (x\geq 3) means “(x) is greater than or equal to (3),” shade to the right Simple, but easy to overlook..
A solid arrow extending from the circle shows that the solutions continue indefinitely in that direction.
5. Write the Result in Interval Notation
The graph can also be expressed using interval notation.
- (x\geq 3) becomes ([3,\infty))
- (x>3) becomes ((3,\infty))
- (x\leq 3) becomes ((-\infty,3])
- (x<3) becomes ((-\infty,3))
A square bracket means the endpoint is included. So a round parenthesis means it is excluded. The symbol (\infty), or infinity, always appears with a parenthesis because infinity is not a specific boundary value Most people skip this — try not to..
Worked Examples
Example 1: Graph (x>-2)
The inequality means that (x) is greater than (-2).
- Mark (-2) on the number line.
- Because the symbol is (>), place an open circle at (-2).
- Shade to the right, toward all values greater than (-2).
The graph looks like this:
←────────────○════════════════════→
-2
The open circle shows that (-2) is excluded. Values such as (-1), (0), and (7) satisfy the inequality, but (-2) and (-3) do not.
In interval notation, the solution is:
[ (-2,\infty) ]
Example 2: Graph (x\leq 5)
This inequality means that (x) is less than or equal to (5) No workaround needed..
- Mark (5) on the number line.
- Since the symbol is (\leq), draw a closed circle at (5).
- Shade to the left.
←════════════════════●────────────→
5
The closed circle indicates that (5) is included. Every number less than (5) also satisfies the inequality.
The interval notation is:
[ (-\infty,5] ]
Example 3: Graph (-4<x\leq 1)
This is a compound inequality. It contains two conditions:
[ -4<x \quad \text{and} \quad x\leq 1 ]
To graph it, mark both boundary points:
- At (-4), use an open circle because (-4) is not included.
- At (1), use a closed circle because (1) is included.
- Shade the region between (-4) and (1).
←────────────○════════════●────────→
-4 1
The solution in interval notation is:
[ (-4,1] ]
Example 4: Solve and Graph (2x-6<8)
Sometimes the first task is to solve the inequality before drawing it.
Start with:
[ 2x-6<8 ]
Add (6) to both sides:
[ 2x<14 ]
Divide both sides by (2):
[ x<7 ]
Now graph (x<7
