📐 Shading Success: Mastering Linear Programming Regions!

Linear programming questions frequently appear in the O-Level Paper 2, and they usually carry a healthy number of marks. **Being able to translate a worded constraint into an inequality and then shade the correct region on a graph is essential** because every optimisation question starts with this skill. If the feasible region is wrong, the final answer will be wrong too!

What is a Feasible Region?

In linear programming, we want to find values of

x

and

y

that satisfy a set of linear inequalities at the same time. **All such points form the *feasible region*.** Interpreting a single inequality is easy; the challenge is combining several of them on one set of axes without mixing up which part to shade. Remember: • **Solid line** for

\le

\ge

(point *on* the line is included). • **Dashed line** for

<

>

(point on the line is *not* included).

Always rearrange each inequality into the form $y \ge mx + c$ or $y \le mx + c$ before drawing, unless it is a vertical line.
Test a simple point, usually $(0,0)$ , to decide which half-plane to shade.
Label the **feasible region** clearly; examiners often award a mark just for a neat, correctly-shaded diagram.

Worked Example: Smoothie Stand Profit

Problem: A school fair sells two types of smoothies—Berry (

x

cups) and Tropical (

y

cups). Each Berry smoothie needs

200\,\text{ml}

of juice and

1

scoop of yoghurt. Each Tropical needs

150\,\text{ml}

of juice and

2

scoops of yoghurt. The stall has

6000\,\text{ml}

of juice and

60

scoops of yoghurt. Write the inequalities that define the feasible region and sketch it. Solution (step-by-step): 1. Juice constraint:

200x + 150y \le 6000

which simplifies to

4x + 3y \le 120

. 2. Yoghurt constraint:

1x + 2y \le 60

. 3. Non-negativity:

x \ge 0,\; y \ge 0

. 4. Draw the lines

4x + 3y = 120

and

x + 2y = 60

on the same axes, using solid lines because of

\le

. 5. For each line, test

(0,0)

: it satisfies both inequalities, so shade the region **towards** the origin for each half-plane. 6. The intersection of all shaded areas (including the first quadrant) is the **feasible region**. Mark it clearly and label the corner points for later optimisation.

By practising the translation from words to inequalities and careful shading, you build a rock-solid foundation for every linear programming question. **Grab graph paper, invent scenarios, and keep drawing those regions**—accuracy comes with repetition, and so do marks!