Linear Inequality – Graphical method, Linear programming

CONTENT

• Linear Inequalities in Two Variables by Graphical Method.
• Graphical Solution of Simultaneous Linear Inequalities in Two Variables.
• Linear Programming

GRAPHICAL SOLUTION OF INEQUALITIES IN TWO VARIABLES

A straight line has the general equation ax+by+c=0, where a,b and c are real numbers.

The line ax + by + c =0 partitions the x-y plane into two regions

SIMULTANEOUS INEQUALITIES

The set of simultaneous inequalities in two variables can be found from the intersection of the areas representing the inequalities.

Evaluation

Show the regions which represent the set of solution of

1)   2y ≤ x + 8,  x + 2y + 4 ≥ 0,  x ≤ 2y + 12

2)   y ≥ 0,  x + 2y ≤ 4,  -x + 2y ≤ 11,  -2x + 5y ≤ 10

LINEAR PROGRAMMING

The linear function z = ax + by is called the objective function while the given set of the inequalities are called the constraint linear programming attempts to maximize or minimize an objective function under the set of given constraints.

Evaluation

1) Show graphically the region represented by the inequalities (a) y ≥ 4x² + 11x – 3   (b) y ≥ 6x² – x – 2

2) Show graphically the region R which satisfies the set of inequalities: 2x + 3y ≤ 26,  x + 2y ≤ 16,  x  ≥ 0, y ≤ 0.

General Evaluation

1. show the region R which satisfies the following simultaneous inequalities

y + x ≤ 3,   y+ x ≥ 1,  y –  x  ≤  1,   x ≥ 0,  y ≥ 0.

• show the region R which satisfies simultaneously 2x + y ≤ 7, 3x – 4y ≥ – 6,   x ≥ 0,  y ≥ 0.

3.    3x² + 7x – 3 = 0       solve using formula method

      4.   Using completing the square and formula method solve 3x² – 12x + 10 = 0

      5.   Solve the following exponential equations (a) 2^2x – 6(2^x) + 8 = 0 (b) 2^2x+1  – 5 (2^x) + 2 = 0

       6.   Janet buys p sweet and q marbles. The sweets cost ₦5 each and the marbles cost ₦6 each. Janet has ₦90.     

             She wants to share the sweets with her friends, so she needs at least 5sweets, she needs more than 4 marbles  

              to be able to join in the game. (a) Write down three inequalities connecting p and q (b) Draw the graph to show 

              their inequalities (c) What is the highest number of sweets she can buy? (d) What is the highest number of 

            marbles she can buy?

Reading Assignment : F/maths Project 1 pg 113 – 119 Exercise 8c Q1, 16 and 17

                              WEEKEND ASSIGNMENT

1) Find the range of x for which │2x – 1│> 3

      (a) 1< x < 3/2  b) -3/2 < x < -1 c) -3/2 < x < 1 d) x > 3/2 and x < -1

2) Find the range of the value that satisfies the inequality x² + 3x – 18 < 0

     (a) -3 < x < 6 (b)-3 > x <6 (c)-6 >x >3 (d)-6 >x < 3 (e)-6 < x <3

3) Find the range of values of x for which 2x² – 5x + 2 ≥ 0

      (a) -2 < x < -½ (b) ½  < x < 2 (c) x < -½ or x ≥ -2 (d) x ≤ ½ or x ≥ 2

4) Find the range of values of y which satisfies the inequality 2y – 1 < 3 and 2 – y ≤ 5

      (a) – 3 ≤ y ≤ 1  (b) – 2 ≤y ≤ 3 (c) -3≤ y ≤ 4 (d) -3 ≤ y ≤ 2

5) Find the range of values of x  for which 1/x + 3 < 2x is satisfy

     (a) – 3 < x < 5/2  (b) x < -3 and x > -5/2  (c) x < 1 and x < ½

                                THEORY

1) Illustrate graphically the set P of all points ( x, y) which satisfy simultaneously the following inequalities:

       2y ≤ x + 8,    x + 2y + 4 ≥ 0,    3x  ≤ 2y + 12. Using your diagram, calculate on the set P the maximum values

       of  (i) x    (ii) y    (iii) 12x + 5y

2) Determine the values of x satisfying |x + 3|  ≥   8