Graphing Inequalities on a Number Line - Sixth Grade Mathematics

Graphing inequalities on a number line visualizes infinite solution sets, using open or closed boundary circles and directional shading to display all satisfying values.

The Anatomy of an Inequality Graph

Because an inequality possesses infinitely many solutions, we cannot represent its solutions with individual points. We represent the solution set by shading an entire section of a number line.

The Two Graphical Components:
1. The Boundary Circle:  Placed at the milestone number.
   - OPEN CIRCLE   (o) : Used for strict inequalities (< or >).
                         Indicates the boundary number is NOT included!
   - CLOSED CIRCLE (*) : Used for inclusive inequalities (<= or >=).
                         Indicates the boundary number IS included!

2. The Directional Arrow:
   - Shade to the RIGHT (-->) for greater than (> or >=).
   - Shade to the LEFT  (<--) for less than (< or <=).

The Four Standard Graph Configurations

Mastering inequality graphing requires recognizing the four visual forms.

+-------------------------------------------------------------------------+
|                  THE FOUR GRAPHICAL CONFIGURATIONS                      |
+---------------+---------------+-----------------------------------------+
| Inequality    | Circle Type   | Shading Direction & Number Line Graphic |
+---------------+---------------+-----------------------------------------+
| x > 3         | OPEN (o)      | Shade RIGHT:                            |
|               | (3 excluded)  | <---|---|---|---o=====================> |
|               |               |     0   1   2   3   4   5   6           |
+---------------+---------------+-----------------------------------------+
| x < 3         | OPEN (o)      | Shade LEFT:                             |
|               | (3 excluded)  | <===================o---|---|---|-----> |
|               |               |     0   1   2   3   4   5   6           |
+---------------+---------------+-----------------------------------------+
| x >= 3        | CLOSED (*)    | Shade RIGHT:                            |
|               | (3 included)  | <---|---|---|---*=====================> |
|               |               |     0   1   2   3   4   5   6           |
+---------------+---------------+-----------------------------------------+
| x <= 3        | CLOSED (*)    | Shade LEFT:                             |
|               | (3 included)  | <===================*---|---|---|-----> |
|               |               |     0   1   2   3   4   5   6           |
+---------------+---------------+-----------------------------------------+

Testing a Point to Verify Shading Direction

A foolproof method to ensure you shaded in the correct direction is the "Test Point Verification": Step 1: Pick any number located clearly inside your shaded region. Step 2: Substitute that test number into the original inequality. Step 3: If the inequality evaluates to TRUE, your shading direction is verified. If FALSE, your shading direction is backwards!

Verification Example for x > 3:
Pick test point in shaded region: x = 5.
Substitute into inequality: 5 > 3. (TRUE!) Shading is correct.
Pick test point in unshaded region: x = 0.
Substitute into inequality: 0 > 3. (FALSE!) Correctly unshaded.

When the Variable is on the Right

When an inequality is written with the variable on the right (such as 4 < x), students often make the mistake of looking at the less-than symbol and shading left. Always read the inequality starting from the variable: 4 < x means "x is greater than 4" (x > 4). You must shade to the RIGHT of 4 with an open circle!

Exercises

Exercise 1. Describe the circle type (open or closed) and shading direction (left or right) for each inequality: (a) x >= -2; (b) y < 5; (c) m <= 0; (d) w > -7.

Exercise 2. Write the algebraic inequality represented by a number line with a closed circle at -4 and shading extending to the right.

Exercise 3. A number line graph has an open circle at 6 with shading extending to the left. Is 6 a solution to the graphed inequality? Is 5.9 a solution?

Solutions

Solution to Exercise 1. For (a) x >= -2: inequality includes "or equal to", so use a Closed circle at -2; greater than means shade to the Right. For (b) y < 5: strict inequality, so use an Open circle at 5; less than means shade to the Left. For (c) m <= 0: includes equality, so use a Closed circle at 0; less than means shade to the Left. For (d) w > -7: strict inequality, so use an Open circle at -7; greater than means shade to the Right.

Solution to Exercise 2. A closed circle at -4 indicates that -4 is included (using <= or >=). Shading extending to the right indicates values greater than -4. Therefore, the algebraic inequality is x >= -4.

Solution to Exercise 3. The open circle at 6 explicitly indicates that 6 itself is excluded from the solution set; therefore, 6 is NOT a solution. However, because shading extends to the left (all numbers less than 6), 5.9 lies inside the shaded region (since 5.9 < 6); therefore, 5.9 IS a solution.