Crucial ECZ Exam Convention
Pay close attention to instructions in ECZ Paper 1 and Paper 2! The standard examination question reads: "By shading the UNWANTED region, show the region R defined by the inequalities..." If you shade the required region instead, you lose all graphing marks!
Table of Contents
1. Inequalities in One Variable
Solving an inequality is identical to solving a linear equation, with one golden exception:
The Negative Multiplication / Division Rule
$$\text{If } -ax \le b \implies x \ge -\frac{b}{a}$$
Whenever you multiply or divide both sides of an inequality by a negative number, you MUST reverse the direction of the inequality sign!
Worked Example: Solve $3 - 2x < 11$.
Step 1: Subtract 3 from both sides: $-2x < 11 - 3 \implies -2x < 8$.
Step 2: Divide both sides by $-2$ and reverse the sign:
$$x > \frac{8}{-2} \implies x > -4$$
Step 1: Subtract 3 from both sides: $-2x < 11 - 3 \implies -2x < 8$.
Step 2: Divide both sides by $-2$ and reverse the sign:
$$x > \frac{8}{-2} \implies x > -4$$
2. Graphing on a Number Line
- Strict Inequalities ($<$ and $>$): Use an Open Circle ($\circ$) to indicate that the boundary endpoint is NOT included in the solution set.
- Inclusive Inequalities ($\le$ and $\ge$): Use a Solid Closed Dot ($\bullet$) to indicate that the boundary value IS included.
- Compound Inequalities: For $-3 \le x < 4$, place a solid dot at $-3$, an open circle at $4$, and connect them with a thick horizontal bar.
3. Two Variables & Boundary Lines
When graphing an inequality such as $2x + y \le 6$ on Cartesian axes:
- Plot the Boundary Line: Replace the inequality sign with an equals sign ($2x + y = 6$). Find the two intercepts:
- When $x = 0$, $y = 6 \implies (0, 6)$.
- When $y = 0$, $2x = 6 \implies x = 3 \implies (3, 0)$.
- Choose Line Type:
- Use a Solid Line if the inequality includes $\le$ or $\ge$.
- Use a Broken / Dashed Line if the inequality is strict ($<$ or $>$).
4. Shading the Unwanted Region
To determine which side of the boundary line to shade:
- Pick a test point not on the line (the origin $(0,0)$ is usually easiest).
- Substitute $(0,0)$ into the inequality: $2(0) + (0) \le 6 \implies 0 \le 6$. This is TRUE.
- Since $(0,0)$ is a valid solution, the region containing $(0,0)$ is the wanted region.
- ECZ Shading Action: Shade the opposite side (the UNWANTED false region), leaving the true region clean!
Ruler & Pencil Precision
Always use a sharp HB pencil and a clear transparent ruler to draw neat parallel hatching lines for unwanted regions. Label the clean feasible region with a prominent capital letter $R$.
Frequently Asked Questions
Multiplying or dividing by a negative number changes the direction of numbers on the number line. For example, $2 < 5$, but multiplying by $-1$ gives $-2 > -5$. Thus, $-3x \le 12$ becomes $x \ge -4$.