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Linear Inequalities in Two Variables · Form 4

Linear Inequalities in Two Variables: Worked Examples (Medium)

This set moves to systems of inequalities: writing all the constraints for a situation, listing the integer points inside a region, and recovering an inequality from a boundary line drawn through two points. It suits students who are steady on single inequalities and ready to combine them.

Written by the spmmath.com.my editorial team.

Worked example 1

A student buys x pens and y notebooks. Each pen costs RM2 and each notebook costs RM5.

She can spend at most RM30, and she wants at least 3 pens. Write a system of inequalities to represent this situation.

  1. Total cost = 2x + 5y, and "at most RM30" gives 2x + 5y ≤ 30.
  2. "At least 3 pens" gives x ≥ 3.
  3. The number of notebooks cannot be negative, so y ≥ 0.
  4. Collect all constraints into one system.

Worked example 2

List all integer points (x, y) with x ≥ 0 and y ≥ 0 that satisfy both x + y ≤ 3 and y ≤ x.

  1. The constraints are x ≥ 0, y ≥ 0, x + y ≤ 3 and y ≤ x. Test each whole-number x in turn.
  2. x = 0: y ≤ x gives y ≤ 0, so y = 0 → (0, 0).
  3. x = 1: y ≤ 1 and x + y ≤ 3 allow y = 0, 1 → (1, 0), (1, 1).
  4. x = 2: y ≤ 2 but x + y ≤ 3 limits y ≤ 1, so y = 0, 1 → (2, 0), (2, 1).
  5. x = 3: x + y ≤ 3 forces y ≤ 0, so y = 0 → (3, 0).

Worked example 3

A boundary line passes through the points (0, 2) and (4, 0). The shaded region lies below this line, and the line itself is included.

Write the inequality that describes the region.

  1. Gradient m = (0 − 2) ÷ (4 − 0) = −2 ÷ 4 = −½.
  2. The y-intercept is 2, so the line is y = −½x + 2.
  3. Multiply by 2 to clear the fraction: 2y = −x + 4, i.e. x + 2y = 4.
  4. Below and included means ≤; test the origin (0, 0): 0 + 0 = 0 ≤ 4 is true.
  5. The origin is below the line and satisfies it, so the sign is correct.

Worked example 4

A workshop makes x tables and y chairs in a day. Each table needs 4 hours and each chair needs 2 hours of work, and there are at most 40 working hours available in a day.

The workshop must make at least 5 chairs. Write all the inequalities that represent this situation (including x ≥ 0).

  1. Total working hours: 4 hours per table and 2 hours per chair, at most 40 hours → 4x + 2y ≤ 40.
  2. At least 5 chairs → y ≥ 5.
  3. The numbers made cannot be negative → x ≥ 0 and y ≥ 0.
  4. So the system is 4x + 2y ≤ 40, y ≥ 5, x ≥ 0, y ≥ 0.

Worked example 5

List all the integer points (x, y) that satisfy all of these inequalities: x ≥ 0, y ≥ 0, x + 2y ≤ 4 and x ≥ y.

  1. Test y = 0: x + 0 ≤ 4 and x ≥ 0, so x = 0, 1, 2, 3, 4 → (0,0), (1,0), (2,0), (3,0), (4,0).
  2. Test y = 1: x + 2 ≤ 4 gives x ≤ 2, and x ≥ 1, so x = 1, 2 → (1,1), (2,1).
  3. Test y = 2: x + 4 ≤ 4 gives x ≤ 0, but x ≥ y = 2 fails, so no points.
  4. Collect all valid points.

Worked example 6

A straight line passes through the points (0, −3) and (3, 3). The region above the line, not including the line itself, is required.

Write the inequality that represents this region.

  1. Gradient m = (3 − (−3)) ÷ (3 − 0) = 6 ÷ 3 = 2.
  2. The line cuts the y-axis at −3, so its equation is y = 2x − 3.
  3. 'Above the line, not including the line' means y is strictly greater than the line's value.
  4. The inequality is y > 2x − 3.

Worked example 7

List all integer points (x, y) with x ≥ 0 and y ≥ 0 that satisfy both 2x + y ≤ 6 and y ≥ 1.

  1. For x = 0: y ≥ 1 and y ≤ 6, so y = 1, 2, 3, 4, 5, 6
  2. For x = 1: y ≥ 1 and 2 + y ≤ 6 so y ≤ 4, so y = 1, 2, 3, 4
  3. For x = 2: y ≥ 1 and 4 + y ≤ 6 so y ≤ 2, so y = 1, 2
  4. For x = 3: 6 + y ≤ 6 means y ≤ 0, which contradicts y ≥ 1, so no points
  5. Collect all the points found

Worked example 8

A boundary line passes through the points (0, 4) and (2, 0). The shaded region lies above this line, and the line itself is not included.

Write the inequality that describes the region.

  1. Gradient = (0 − 4) / (2 − 0) = −4/2 = −2
  2. Using the y-intercept 4, the line is y = −2x + 4
  3. 'Above the line, not included' means y > −2x + 4
  4. Rearranged: 2x + y > 4

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

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Frequently asked questions

How should I approach medium-level questions that combine two or more linear inequalities?

Draw every boundary line on the same axes first, using the correct scale, then shade each inequality's region and identify the feasible region where all shadings overlap. Label the feasible region clearly, as some questions ask you to read a maximum or minimum value from it.

Working line by line avoids mixing up regions.

What's a common mistake when choosing the scale for the axes in these questions?

Students sometimes pick a scale too small to fit all the given inequalities on the provided grid, forcing them to redraw everything. Before plotting, check the largest intercept among all the inequalities and choose a scale that comfortably fits it, using the graph paper provided rather than guessing.

How do I check that a point actually lies in the feasible region?

Substitute the point's coordinates into every inequality in the question, not just one of them, a point only lies in the feasible region if it satisfies all the inequalities simultaneously. This substitution check is also the fastest way to verify your shaded diagram before moving on to the next part.

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