Types of Compound Inequalities

Introduction to types of Compound inequalities:

Compound Inequalities are is defined as the terms of condition availability in the particular inequalities conditions are satisfied in between two or more simple inequalities joined by the terms 'and' or 'or'.

Types of compound inequalities are

There are three types of inequalities

1.  And - this is the inequality that has separated the two inequalities.

2.  Or   - this is the inequality that has separated the two inequalities.

3.  a < bx + c < d – this is the single inequality form.

And, or Types of Compound Inequalities with Examples:
And compound inequalities examples:

This is like an intersection of the particular two solutions. It is easy to solve. To find the intersection of particular two sets.

Ex:1    2x + 2 < 8 and 3x – 4 > -4

Sol:          2x < 8 – 2          3x > -4 + 4

2x < 6                3x > 0

x < 3                 x > 0          

The intersection and the inequalities of the solution is (x | 0 < x < 3)

Ex: 2 Solve    4x +5 > 1 and 3x – 4 < -1

Solution:4x > 1 – 5 and 3x < -1 + 4

4x > -4     and 3x < 3

x > -1      and   x < 1        

The intersection and the inequalities of the solution is ( x| -1 < x < 1)

Or compound inequalities :

This type is like a union of the particular two solutions inequality. It is easy to solve. To find the union of particular two points.

Ex: 2x - 3 > 7  or  3x – 2 < -5

2x > 4     or   3x < -3

X > 2    or    x < -1   

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Types of Compound Inequalities Form:

Single compound inequalities form:


This type inequalities are the as long as the variable has to be appeared only in the middle part of the variable in their inequality.


Ex: -2 < 5x -7 < 3


Sol:Given       -2 < 5x – 7 < 3                        add the 7 both side of 3 parts


5 < 5x < 10                            divide each of the 3 parts by 5


1 < x < 2


The answer is(x | 1 < x < 2)

Ordinal Variable Example

Introduction ordinal variable example:

          Ordinal variable is a variable which is mainly used to give order or to rank the particular data. There can be no any arithmetic operation performed on an ordinal variable like a normal mathematical variables. The important use of Ordinal variables is comparison of data to rank them according to its performance. The ranking given using ordinal variables is also known as level of data.

Ordinal Variable Example Explanation:

      Ordinal variable is sometimes known as discrete data. Our ranking given to data by ordinal variable must be in meaningful way. For example,

• Agree / disagree
• Maximum/ minimum

 

I am planning to write more post on how to solve andgraph inequalities, solving systems of linear equations by graphing. Keep checking my blog.

          Ordinal variable is plays a vital role in surveying and we can do logical operations on ordinal variables to compare the data. 

Ordinal Variable Example:

 Example 1:

We can give the ranking order for the ordinal variable as follows,

1 = Excellent

2 = Good

3 = Satisfactory

4 = Low

5 = Poor

By the above exposed values, the data are analyzed.

For example, we are taking the labor’s performance on the work wage. This can be given as; if the work of the labor is above 10 points means then the ranking will be 1. If the work of the labor is above 8 points means, then the ranking will be 2. And If work level is above 5 means the rank will be 3. Likewise below 5 means low and below 3 means very poor. This is what the ordinal variable is performed.

Representing Ordinal variable example in the table format,

1	2	3	4	5
Excellent	good	ok	low	Poor

The performance level of the labors can be calculated by using one of the five stages. This ordinal variable table will be more helpful while analyzing individual performance also

Ordinal variable Example 2:

We can also use ordinal variable in medical research by taking the survey of patients according to our requirement.

Introduction to parallel and perpendicular lines worksheet

Introduction to parallel and perpendicular lines worksheet
The equation of a straight line is an algebraic condition which is satisfied by every point on it.

The equation of a straight line is an algebraic condition which is satisfied by every point on it.

The various forms of straight lines

The equation of a straight line in various forms

Normal form of an equation is x cos a + y sin ß = p
Intercept form is x / a + y / b = 1
Symmetric form (x - x ) :cos ? = (y - y) :sin ?
General form of an equation ax + by +c = 0 (a2 + b2 ? 0 )
Parametic form x = x1 + r cos ? ; y= y1 + r sin ?

Parallel and Perpendicular Lines Worksheet : Slope

The slope of a straight line is represented in various forms from different forms of the equations

If the given vertices are (x1,y1) and (x2 ,y2 ) then the slope m is taken as y2 - y1 / x2 - x1

The slope of an equation can be represented as m = dy / dx where the derivatives are applied

If the line is in the form of a tanget equation y = mx +c then the slope will be equal to 'm'.

If the straight line makes an angle  theta with x axes then tan theta is called the slope.

If theta is acute, then tan theta is positive.

If theta is obtuse, then tan theta is negative.

If two lines are parallel, then they have same slope

If two lines are perpendicular, then m2 = -1/m1

where m2 is slope of second line

          m1 is slope of first line.

The slope concept for parallel and perpendicular lines are useful in  solving problems in worksheet.

Problems on Parallel and Perpendicular Lines Worksheet

Find the equation of line passing through ( 0,0) and parallel to   y = 3x + 2

Solution:  Given equation  is    y  = 3x +2

                slope of line  = m1  = 3

 As the lines where parallel  m1 = m2  = 3

Equation of line passing through (0,0)

is y = m2 x + c

As the line passes through origin , replace x and y values by 0,0 to find value of c

0 = 3(0) +c

c = 0

So equation of line parallel to y = 3x + 2 is   y = 3x

2) Find equation of line passing through (1,1) and perpendicular to y  = x + 3

Solution)  Given equation :  y = x + 3

           m1 =      Slope of line =  1

As lines where perpendicular , product of their slopes are equal to -1

                                               m1 *m2 = -1                   ( * represents multiplication symbol)

                                                   m2 = -1

Equation of line :   y  = m2 x + c

                           y = -x  +c

as the equation passes through (1,1) replace x and y by 1 , 1

                   1 = -1 + c

                     c = 2

Equation of required line is   y  = m2 x  + c

                                         y = -x + 2

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Introduction to angles of a pentagon

Introduction to angles of a pentagon:

              A pentagon is a closed two dimensional figure that is the union of line segments in a plane. A pentagon has the five sides and five angles.

              Pentagon can be classified as

•  Convex pentagon
• Concave pentagon

             For  a given pentagon, we can measure the following angles of Pentagon.

•  Interior angle
• Exterior angle

Interior Angles of a Pentagon:

             An angle located within a closed figure or the angle formed inside a polygon by two adjacent sides is referred to as an Interior angle.   

             For finding the interior angles of a pentagon, start with one vertex and join it to all other vertices to form three triangles inside the pentagon.

            A pentagon has 5 sides and 5 vertices and three triangles are formed by connecting the vertices, i.e., number of triangles inside a pentagon is two less than the sides of the pentagon. So, the sum of all interior angles in a polygon is given by,

                                      Sum of all interior angles = 180 ( n – 2 )

                     For a pentagon, number of sides n = 5 and the sum of all interior angles are

                                                                                    = 180 ( n – 2 )

                                                                                    = 180 ( 5 – 2 )

                                                                                    = 180 * 3

                                                                                    = 540 degrees.

                     Each interior angle of a pentagon  = 540 / 5

                                                                                   = 108 degrees.

Ex 1: Find the number of sides of pentagon if the sum of interior angles of pentagon is 540 degrees.

Sol: Step 1: Sum of interior angles of pentagon = ( n - 2) x 180°, n is number of sides.

                        540°   =  ( n - 2 ) x 180°

         Step 2:  Divide by 180° to each side.

                        540°/180° = ( n - 2 ) x 180° / 180°

                             3           =  n - 2

         Step 3: Add 2 to each side.

                            3 + 2      = n - 2 + 2

                                5         =  n

Therefore, nmber of sides for pentagon is five.

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Exterior Angles of a Pentagon:

            An exterior angle of a pentagon is an angle formed by two sides of the pentagon which normally shares a common vertex. In a vertex, two exterior angles can be formed.

           In each vertex the angle formed by extending a side is equal to 180 degree

                                       Supplementary angle  = interior angle + exterior angle

           For a n sided polygon,

                                    Sum of all exterior angles, = 180 n – 180 ( n - 2 )

                                                                                     = 180 n – 180 n + ( - 180  x  – 2 )

                                                                                     = 0 + 360

                                                                                     = 360 degrees. 

                      So,The sum of exterior angles of a pentagon = 360 degrees

                      In a pentagon, for 5 vertices 5 exterior angles can be formed.      

                       Each exterior angle of a pentagon = 360 / 5 = 72 degrees.