INEQUALITIES > means left side is greater than right side < means left side is less than right side ≤ means left side is less than or equal to right side ≥ means left side is greater than or equal to right side
Example1: Solve: 2x^{2} + 3x – 5 < 0
y = 2x^{2} + 3x – 5 = (2x + 5)(x  1) = 0
Roots of the equation: x_{1} =  5/2 and x_{2} = 1
Example 2:
2x^{4} – x^{3} – 47 x^{2} + x + 45 > 0
Factor: y = (2x + 9)(x  5)(x  1)(x + 1) = 0
Roots: x_{1} =  4.5, x_{2} = 5, x_{3} = 1, x_{4} = 1

Example 3:
Solve the following system of inequalities: x^{2} + y^{2} < 64 ∩ 2x^{2} + 5y ≤ 0 ∩ x + 3y > 12
x^{2} + y^{2} < 64
2x^{2} + 5y ≤ 0
x + 3y > 12
SOLUTION:
Step 1
Graph: x^{2} + y^{2} = 64
Since x^{2} + y^{2} less than 64, a dashed line is used in the drawing.
Graph: 2x^{2} + 5y = 0
Since 2x^{2} + 5y equal and less than zero, a solid line is used in the drawing.
Graph: x + 3y = 12
Since x + 3y greater than 12, a dashed line is used in the drawing.
Step 2
To decide the inside or outside of the circle is true, take any point inside or outside of the circle. Let us take: (0 , 0) Substitute x = 0 and y = 0 0 + 0 < 64 Therefore inside of the circle is true.
To decide for the parabola: 2x^{2} + 5y ≤ 0 Take any point above or below the parabola. Let us take: (0 , 1) Substitute x = 0 and y = 1 0 + 5 NOT less than or equal to zero Therefore above of the parabola is false, below of the parabola is true.
To decide for the line: x + 3y > 12 Take any point below or above the line Let us take: (0 , 0) Substitute x = 0 and y = 0 0 + 0 > 12 Therefore above the line is true.
Step 3
Shade the area that is true for all three inequalities.
The graph is given below:
