Inequalities and linear programming

What you will learn

• 8.1 Compound linear inequalities with "and" or "or"; triangle inequality problems.
• 8.2 Quadratic inequalities by the graphical method.
• NF 8.3 Quadratic inequalities by the algebraic method.
• NF 8.4 Graphs of linear inequalities in two unknowns.
• NF 8.5 Systems of linear inequalities in two unknowns.
• NF 8.6 Linear programming.

You need first: Linear inequalities in one unknown, Quadratic equations in one unknown, Equations of straight lines

Key terms

Standard steps

Compound inequalities

1. Solve each part separately.
2. Draw both on one number line.
3. "and": the overlap only. "or": everything covered by either part.

Triangle inequality

The sum of any two sides is greater than the third side.

1. Write all three inequalities: a + b > c, a + c > b and b + c > a.
2. Solve each one for the unknown side.
3. Combine them: the unknown side lies between the difference and the sum of the other two sides.

Quadratic inequality

1. Move everything to one side and factorise (or find roots with the formula).
2. Sketch the parabola through the roots (a > 0 opens up).
3. "< 0" means below the x-axis: between the roots.
4. "> 0" means above: outside the roots.

y = (x + 2)(x − 3)
  \         /
---•-------•---      x² − x − 6 < 0   →   −2 < x < 3   (below)
   −2  \_/  3        x² − x − 6 > 0   →   x < −2 or x > 3 (above)

Linear programming (NF)

1. Define the unknowns: "Let x be the number of …".
2. Write the constraints as inequalities, including x ≥ 0, y ≥ 0 and "integers" if needed.
3. Draw each boundary line and find the feasible region, where every constraint is true. Follow the question's shading convention.
4. Find the vertices of the region (solve pairs of boundary lines simultaneously).
5. Test the objective function at each vertex. The maximum or minimum is at a vertex.
6. If the answer must be an integer and a vertex isn't, test the nearby integer points inside the region.

Why a vertex: the objective function is a straight line sliding across the region; it last touches the region at a corner.

Worked examples

2x − 1 < 5   →  x < 3
3 − x ≤ 4    →  x ≥ −1
"and":  −1 ≤ x < 3

vertices: (0, 0), (12, 0), (8, 4), (0, 8)
          [(8, 4) from x + y = 12 and x + 2y = 16]
P:          0,      360,     400,     320
maximum P = 400 at x = 8, y = 4

Let x be the number of chairs and y the number of tables.
constraints:  x + 3y ≤ 15          cutting
              2x + 2y ≤ 14, i.e. x + y ≤ 7      finishing
              x ≥ 0, y ≥ 0, x and y integers
objective:    P = 50x + 90y

vertices: (0, 0), (7, 0), (3, 4), (0, 5)
          [(3, 4) from x + 3y = 15 and x + y = 7]
P:          0,     350,    510,    450
greatest profit $510: 3 chairs and 4 tables

4 + 9 > x   →  x < 13
4 + x > 9   →  x > 5
9 + x > 4   →  always true
5 < x < 13

Common mistakes

• Using "and" when the question says "or" (and the reverse).
• For a quadratic inequality, giving "x < 3 and x > −2" when the parabola opens down (always sketch).
• Dividing a quadratic inequality by x (x could be negative).
• Testing points that are not vertices, or missing a vertex on an axis.
• Forgetting the integer condition in a real-life LP problem.

How the DSE tests it

• A(1): compound inequalities with "and" or "or", then the number of integers that satisfy them.
• Paper 2: quadratic inequalities; which region represents a system of inequalities; NF the maximum of an objective function on a given region.
• Section B: occasionally a full LP word problem.

Questions you might not ask in class

More Form 5 maths topics