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:   2x2 + 3x 5 < 0

 

y = 2x2 + 3x 5 = (2x + 5)(x - 1) = 0

 

Roots of the equation:  x1 = - 5/2 and x2 = 1

 

 

x < - 5/2

x = -5/2

-5/2 < x < 1

x = 1

 1 < x

Sign of (2x + 5)

-

0

+

+

+

Sign of (x - 1)

-

-

-

0

+

Sign of (2x + 5)(x - 1)

+

0

-

0

+

 

Polynomial: 2x2 + 3x 5 < 0  in this interval:   -5/2 < x < 1,

 

 

 

 

Example 2:

 

2x4 x3 47 x2 + x + 45 > 0

 

Factor: y = (2x + 9)(x - 5)(x - 1)(x + 1) = 0

 

Roots:  x1 = - 4.5,    x2 = 5,    x3 = 1,    x4 = -1

 

From the graph, Polynomial:

 

 2x4 x3 47 x2 + x + 45 > 0

 

in these intervals:    x < -4.5,     -1 < x < 1,     5 < x

 

 

Example 3:

 

Solve the following system of inequalities:  x2 + y2 < 64 ∩ 2x2 + 5y ≤ 0 ∩ x + 3y > -12

 

x2 + y2 < 64

 

2x2 + 5y ≤ 0

 

x + 3y > -12

 

 

SOLUTION:

 

Step 1

 

Graph:  x2 + y2 = 64

 

Since x2 + y2 less than 64, a dashed line is used in the drawing.

 

Graph:  2x2 + 5y = 0

 

Since 2x2 + 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:  2x2 + 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: