Ronald Hui Tak Sun Secondary School HKDSE Mathematics Ronald Hui Tak Sun Secondary School

Schedule UT1 Homework collection Monday (16 Nov) Return Next next week Ronald HUI 22 October 2015

Form 5 Mathematics Chapter 4 Linear Programming Form 5 Mathematics Chapter 4

We usually represent the solutions of a linear inequality in two unknowns graphically by shading the relevant half-plane. Note that when the boundary is not a part of the solutions, it is drawn as a dotted line. Solutions of x + y > 4 Solutions of x + y < 4

We usually represent the solutions of a linear inequality in two unknowns graphically by shading the relevant half-plane. Note that when the boundary is a part of the solutions, it is drawn as a solid line. Solutions of x + y  4 Solutions of x + y  4

Step 1 Solve the inequality 2x + y  0 graphically. Draw the boundary with a dotted/solid line according to the inequality sign. Inequality sign Boundary ‘’ or ‘’ solid line ‘>’ or ‘<‘ dotted line

Step 1 Solve the inequality 2x + y  0 graphically. Draw the boundary with a dotted/solid line according to the inequality sign. Draw the solid line 2x + y = 0.

Step 2 Solve the inequality 2x + y  0 graphically. Choose a test point and check which half-plane represents the solutions of the inequality. Choose (1, 1) as the test point. When x = 1 and y = 1, 2x + y = 2(1) + 1 = 3  0 We can choose any point that does not lie on the boundary as the test point. ∴ The half-plane containing the test point (1, 1) together with the boundary represents the solutions of the inequality.

Graphical representation of 2x + y  0 Solve the inequality 2x + y  0 graphically. Step 3 Shade the half-plane found in step 2. Graphical representation of 2x + y  0

Solving Systems of Linear Inequalities in Two Unknowns Graphically

To solve a system of linear inequalities in x and y, Systems of Linear Inequalities in Two Unknowns î í ì ³ - £ + 3 2 1 y x Example: (i) î í ì £ + ³ 7 5 4 y x (ii) To solve a system of linear inequalities in x and y,

Systems of Linear Inequalities in Two Unknowns î í ì ³ - £ + 3 2 1 y x Example: (i) î í ì £ + ³ 7 5 4 y x (ii) we have to find all ordered pairs (x, y) satisfying all the linear inequalities.

This can be done effectively through the graphical method. Systems of Linear Inequalities in Two Unknowns î í ì ³ - £ + 3 2 1 y x Example: (i) î í ì £ + ³ 7 5 4 y x (ii) This can be done effectively through the graphical method.

I can solve the two linear inequalities graphically. î í ì ³ - £ + 3 2 1 y x Solve the system of inequalities graphically. Solutions of x + y £ 1 Solutions of x – 2y ³ 3 I can solve the two linear inequalities graphically.

But what are the solutions î í ì ³ - £ + 3 2 1 y x Solve the system of inequalities graphically. Solutions of x + y £ 1 Solutions of x – 2y ³ 3 But what are the solutions for ? î í ì ³ - £ + 3 2 1 y x

Solve the system of inequalities graphically. î í ì ³ - £ + 3 2 1 y x Solve the system of inequalities graphically. Solutions of x + y £ 1 Solutions of x – 2y ³ 3 Let me show you the steps.

î í ì ³ - £ + 3 2 1 y x Solve the system of inequalities graphically. Step 1 Draw the graphical representations of the two linear inequalities in two unknowns on the same coordinate plane. Graphical representation of x + y £ 1 Graphical representation of x – 2y ³ 3

Step 2 Find the overlapping region of the two graphical representations. overlapping region All ordered pairs in the overlapping region satisfy both x + y £ 1 and x – 2y ³ 3.

Step 2 Find the overlapping region of the two graphical representations. overlapping region Hence, it represents the solutions of the system of linear inequalities.

Solve the system of inequalities graphically. Usually, we simply add arrows on the boundaries to indicate which half-plane represents the solutions of individual inequalities. î í ì ³ - £ + 3 2 1 y x Solve the system of inequalities graphically. ◄ For each boundary, the direction of arrow indicates the half-plane that represents the solutions of the corresponding inequality.

Then, we shade the overlapping region only. î í ì ³ - £ + 3 2 1 y x Solve the system of inequalities graphically. ◄ For each boundary, the direction of arrow indicates the half-plane that represents the solutions of the corresponding inequality.

î í ì  +  4 y x Solve the system of inequalities graphically. What if the two graphical representations have no overlapping region like this example?

In this case, the system of inequalities has no solutions. î í ì  +  4 y x Solve the system of inequalities graphically. In this case, the system of inequalities has no solutions.

Follow-up question Solve the system of inequalities graphically. ï î í ì ³ £ - > x y 2 1 Draw x = –1 as a dotted line, y = 2 and y = x as solid lines on the same coordinate plane.

Follow-up question Solve the system of inequalities graphically. ï î í ì ³ £ - > x y 2 1 Determine which half-planes represent the solutions of x > –1, y  2 and y  x respectively, and indicate them by arrows.

Follow-up question Solve the system of inequalities graphically. ï î í ì ³ £ - > x y 2 1 Shade the overlapping region of the three half-planes. ◄ The shaded region represents the solutions of the system of inequalities.