System of Inequalities

Inequalities:

A number a > a number b if a - b is positive.

i.e. a > b

Example: 2 > 1 because 2-1 = 1 is positive.

 -1 > 2 is false because -1-2 = -3 is negative.      

Rules for inequalities:

• An equal quantity may be added to, (or subtracted from) both sides of an inequality without changing the inequality.
  4 > -1
  4-3 >-1-3(subtract 3)
  1 > -2 which is also true.
• An equal positive quantity may multiply (or divide) both sides of an inequality without changing the inequality.
  4 > -1
  4 x 2 > -1 x 2(multiply by 2)
  8 > -2 which is also true.
• If both sides of an inequality are multiplied ( or divided by a negative quantity then the inequality is reversed.
  4 > -1
  4 x -3 < -1 x -3(multiply by -3; sign reversed)
  -12 < 3 which is true.

Solving inequalities:

• x-5 = 0 gives x=5
• x-5 > 7
  x-5+5 > 7+5 ( add 5 on both sides)
  x > 12
• 2x-2 < x
  2x-2+2 < x+2 ( add 2 on both sides)
  2x < x+2
  2x-x < x+2-x (subtract x on both sides)
  x<2

Definition of System of Inequalities

• A system of inequalities is a set of two or more inequalities with the same variables, graphed in the same coordinate plane.

Examples of System of Inequalities

The figure below shows the graph of the system of inequality y ≤ x+2

The figure below shows the graph of the system of inequalities 2x - 6y < 12, 3x + 4y < 12, and 4x + 2y ≥ 8. 

Choose the correct graph:

Which of the graphs defines the system of inequalities, x + y < 3, - 5x + 2y ≤ 10, y ≥ - 3?

Solution: 
Step 1: Graph all the three inequalities in the same coordinate plane.

Step 2: The graph of x + y < 3 is the half-plane below the dashed line x + y = 3.
Step 3: The graph of - 5x + 2y ≤ 10 is the half-plane on and below the solid line - 5x + 2y = 10.
Step 4: The graph of y ≥ - 3  is the half-plane on and above the solid line y = - 3.
Step 5: The graph of the system is the intersection of the three half planes as shown in the graph.
Step 6: Graph-1 is the correct answer.