🔢 Drawing the Line: Visualising Inequalities with Confidence!

Understanding how to **represent inequalities on a number line** is a core skill for O Level Mathematics. It turns abstract algebraic statements into clear visual pictures, helping you check solutions quickly in exams and real-life problem-solving.

Key Ideas You Must Know

An inequality shows that one value is **greater than**, **less than**, or **equal to** another. On a number line, we use open or closed circles (or dots) and arrows to display the set of possible values.

Open circle ⭕ for $<$ or $>$ (value NOT included).
Closed dot • for $\le$ or $\ge$ (value included).
Shade/arrow to the left for "smaller than", to the right for "greater than".

Worked Example: Graphing & Solving a Linear Inequality

Problem: Solve

\,-2 < 3x + 1 \le 7\,

and represent the solution on a number line.

Step 1:
$-2 < 3x + 1 \le 7$
Subtract $1$ throughout:
$-3 < 3x \le 6$
Step 2: Divide every part by $3$ :
$-1 < x \le 2$
Step 3: Draw a number line, mark $-1$ with an *open* circle (value not included) and $2$ with a *closed* dot (value included). Shade the region between them.

Practice sketching a variety of inequalities—single, double, and even compound ones—to build speed and accuracy. **Consistent practice turns these quick marks on a line into exam marks on your script!**