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Linear Inequalities in Two Variables · Form 4

Linear Inequalities in Two Variables: Worked Examples (Easier)

This set builds the three starting skills: turning one worded constraint into an inequality, testing whether a point satisfies an inequality, and deciding on a solid or dashed boundary and which side to shade. It suits students meeting linear inequalities in two variables for the first time.

Written by the spmmath.com.my editorial team.

Worked example 1

A hall is set up with x round tables and y long tables. The total number of tables must not be more than 40.

Write an inequality to represent this constraint.

  1. Total number of tables = x + y.
  2. "Not more than 40" means the total is at most 40, so use ≤.
  3. Combine both parts: x + y ≤ 40.

Worked example 2

Determine whether the point (2, 3) satisfies the inequality 2x + y ≤ 8.

  1. Substitute x = 2 and y = 3 into the left side: 2(2) + 3.
  2. Evaluate: 4 + 3 = 7.
  3. Compare with the right side: 7 ≤ 8 is true.
  4. A true statement means the point lies in the region.

Worked example 3

For the inequality y ≥ x + 1, state whether the boundary line y = x + 1 should be drawn solid or dashed, and whether the region above or below the line should be shaded.

  1. The sign is ≥, which includes equality, so the boundary line is solid.
  2. y ≥ x + 1 means y is greater than or equal, pointing to the region above the line.
  3. Test the point (0, 3): 3 ≥ 0 + 1 gives 3 ≥ 1, which is true.
  4. Since (0, 3) is above the line and satisfies the inequality, shade above.

Worked example 4

On a graph, the line y = 2x − 1 is drawn as a dashed line and the region above the line is shaded. Write the inequality that describes the shaded region.

  1. A dashed line means the boundary is not included, so the inequality is strict (< or >).
  2. The region above the line means y is greater than the line's value.
  3. Therefore the inequality is y > 2x − 1.

Worked example 5

The point (3, y) satisfies the inequality x + y ≥ 8. Find the smallest integer value of y.

  1. Substitute x = 3 into the inequality: 3 + y ≥ 8.
  2. Subtract 3 from both sides: y ≥ 5.
  3. The smallest integer that is at least 5 is 5.

Worked example 6

For the inequality x + y ≤ 4, state whether the boundary line should be drawn solid or dashed, and whether the region below or above the line is shaded.

  1. The sign is ≤, which includes 'equal to', so the boundary line is solid.
  2. Rearrange to y ≤ 4 − x; y is less than or equal to the line's value.
  3. 'Less than' means the region below the line is shaded.

Worked example 7

At a charity book sale, x fiction books and y non-fiction books are packed into one box. Each box must contain at least 50 books in total.

Write an inequality to represent this constraint.

  1. Total books in the box = x + y
  2. 'At least 50' means the total must be 50 or more
  3. x + y ≥ 50

Worked example 8

Determine whether the point (5, 1) satisfies the inequality 3x − y < 10.

  1. Substitute x = 5, y = 1 into 3x − y
  2. 3(5) − 1 = 15 − 1 = 14
  3. Since 14 < 10 is false, the point does not satisfy the inequality

Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)

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Frequently asked questions

What's the key skill tested in easy questions on linear inequalities in two variables?

Easy questions check whether you can plot the boundary line of an inequality accurately and shade the correct side. The line is solid for ≤ or ≥ and dashed for < or >, and you test a point (often the origin) in the original inequality to decide which side to shade.

Getting the line type right is an easy mark many students lose.

Why does the origin sometimes not work as a test point?

If the boundary line passes through the origin itself, testing (0, 0) gives no useful information because it lies exactly on the line. In that case, choose any other easy point not on the line, such as (1, 0) or (0, 1), and substitute it into the original inequality to check which side is correct.

Do I need to shade the wanted region or the unwanted (rejected) region?

Read the question's instruction carefully, SPM often asks you to shade the unwanted region, leaving the answer (feasible) region blank and clear. Always write "R" or label the required region on your diagram regardless of which side is shaded, so the marker can see exactly which area you mean.

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