Understanding Inequalities and Number Lines
An inequality expresses a relationship between two quantities using one of four symbols: less than (<), less than or equal to (≤), greater than (>), or greater than or equal to (≥). For example, x < 10 tells us that x can be any number below 10, but not 10 itself.
A number line is a horizontal axis representing all real numbers in order. It extends infinitely in both directions, with zero typically at the centre. Every point on the line corresponds to exactly one real number, allowing us to visualise where solutions lie in relation to other values.
When we graph an inequality on a number line, we mark a critical point and shade the region representing all numbers that satisfy the condition. The shading direction (left or right) depends on whether the inequality points left (<, ≤) or right (>, ≥). The endpoint itself is marked with an open circle for strict inequalities (<, >) or a closed circle for non-strict ones (≤, ≥).
Systems of Inequalities on a Number Line
A system of inequalities requires that multiple conditions hold true simultaneously. For instance, −2 ≤ x < 5 means x must satisfy both x ≥ −2 AND x < 5. The solution set is the overlapping region where all shadings meet.
Graphing compound inequalities involves:
- Solving each inequality independently to identify its critical point
- Plotting each on the same number line, often using different colours or styles
- Identifying the intersection—the region where all shadings overlap
If no overlap exists, the system has no solution. For example, x > 10 and x < 5 cannot both be true, so there is no value of x satisfying both conditions. Conversely, −3 < x ≤ 7 has infinitely many solutions: every number strictly greater than −3 and up to and including 7.
Practical Tips for Number Line Graphing
These common pitfalls will sharpen your number line graphing skills.
- Circle type matters — Open circles (○) represent strict inequalities (<, >), excluding the boundary point itself. Closed circles (●) represent non-strict inequalities (≤, ≥), including the boundary. Confusing these will make your solution set incorrect.
- Shade direction follows the symbol — The inequality symbol points in the direction you shade. x > 3 shades right from 3; x ≤ −1 shades left from −1. Reversing the shading direction inverts your entire solution set.
- Test a point when unsure — Pick a test value inside your shaded region and substitute it into the original inequality. If it satisfies the inequality, you've shaded correctly. This quick check prevents errors, especially with compound systems.
- Handle negatives carefully in systems — When graphing −5 < x ≤ 2, ensure your shading extends from just after −5 (open circle) rightward to 2 (closed circle). Negative numbers still follow the same left-to-right ordering; don't flip the logic.
Practical Example: Solving a Compound Inequality
Suppose we need to graph the system: x < 2 and x ≥ −1.
Step 1: Identify the critical points: −1 and 2.
Step 2: For x ≥ −1, place a closed circle at −1 and shade rightward.
Step 3: For x < 2, place an open circle at 2 and shade leftward.
Step 4: The solution is where both shadings overlap: the region from −1 (included) to 2 (excluded), written as −1 ≤ x < 2.
This interval contains infinitely many values: −1, −0.5, 0, 0.999, 1, 1.5, and so on—everything up to but not including 2. Visually, you'll see a solid line segment on the number line from the closed circle at −1 to the open circle at 2.
Reading Existing Number Line Graphs
Interpreting a shaded number line requires identifying three elements:
- The boundary point: Where the shading begins or ends.
- The circle type: Open (strict) or closed (non-strict).
- The shading direction: Left means < or ≤; right means > or ≥.
For example, if you see a closed circle at 5 with a thick line extending to the right, the inequality is x ≥ 5. If instead the circle is open at 5 with shading to the left, the inequality becomes x < 5.
With multiple shadings, the solution region is where all colours overlap. An empty overlap signals no solution exists.