Graph Systems of Inequalities

Algebra is a subdivision in mathematics in which comprises of infinite number of operations on equations, polynomials, inequalities, radicals, rational numbers, logarithms, etc. Graphing system algebra inequalities is also a part of algebra. It is similar to the graphing of ordinary equations, but here we got to graph the inequality given either greater or less than. In graphing system of inequalities , the area overlapping is the solution set.. The following sections helps us to learn much better about graphing system of inequalities.

Steps to solve system of inequalities:

The steps involved in graphing system of inequalities are as follows,

Consider the inequality y > ax+c
Step 1: Convert the given equation as y =ax+c.
Step 2: Since the given inequality is a function of x, let y =f(x).
Step 3: Therefore f(x) = ax+c.
Step 4: Substitute various values for ‘x’ and find corresponding f(x).
Step 5: Table the values as follows x  &  f(x)  the values of x as -3, -2, -1, 0,1,2,3. and for f(x) their corresponding values.
Step 6: The values in the table are the co-ordinates, graph them.
Step 7: Shade the inequality range above the line, since greater than symbol (>) is given.
Step 8: Shade the inequality range below the line, if less than symbol (>) is given.
Step9: Repeat the same steps for next equation also.
Step 10: Shade the region which is overlapped, which is the solution region for the system of inequalities.


Example problem for graphing inequalities:

Problem 1:

Graph the given system of inequalities and find the solution region,

3x - y < 4

2x + y < 3
Convert the given equation as

y >3x+4
Since the given inequality is a function of x,
Let y =f(x).
Therefore
f(x) = 3x+4.

Substitute various values for ‘x’ and find corresponding f(x).
When x= -3
f(-3) = (-3)3 +4,
-9+4,
-5, therefore the co-ordinates are (-3, -5)

When x= -2
f(-2) = (-2)3 +4,
-6+4,
-2, therefore the co-ordinates are (-2, -2)

When x= -1,
f(-1) = (-1)3 +4,
-3+4,
1, therefore the co-ordinates are (-1, 1)

When x= 0
f(0) = (0)3 +4,
0+4,
4, therefore the co-ordinates are (0, 4)

When x= 1
f(1) = (1)3 +4,
3+4,
7, therefore the co-ordinates are (1, 7)

When x= 2
f(2) = (2)3 +4,
6+4,
10, therefore the co-ordinates are (2, 10)
Tabulate and graph them, and mark the inequality
Following the same procedure for the second equation and graph them. the graph looks like,

On Graphing them combined, we get the solution region as ,

Problem 2:

Graph the system of inequalities

4x +3y <2

2x+y >8

Solution:

Following the above procedure and graphing them, the solution range is as follows,

Radical Rules Math

Radical symbol used to indicate the square root or nth root. Radical of an algebraic group, a concept in algebraic group theory. Radical of a ring, in ring theory, a branch of mathematics, a radical of a ring is an ideal of "bad" elements of the ring. Radical of a module, in the theory of modules, the radical of a module is a component in the theory of structure and classification. Radical of an ideal, an important concept in abstract algebra. The radical symbol is ' √ ' . The cubic root of a can be expressed as `root(3)(a)`   and nth root of x can be expressed as  ` rootn x `                                                                                                                                                                                                   

                                                                                                                                                                                                                   Source Wikipedia.

I like to share this sine rules with you all through my article. 

Radical rules in math:

                    Math radical has product rule, Division rule and exponential rule.
Math Product rules for radical:
              Square root product rule:         `sqrt(ab)`   =  ` sqrta * sqrt b`
              Cube root product rule:          `root3 (ab) `   = ` root3 (a) * root3 (b)`
              n^th root product rule                ` rootn (ab) = rootn (a) * rootn (b)`
Examples for radical product rule math problem 1:
              Multiply the two math radical `sqrt5 * sqrt14`
        Solution:
                            Given radicals`sqrt5 * sqrt14`        
               We know the math radical product rule  `sqrt(ab)`   =  ` sqrta * sqrt b` 
                                     So, `sqrt5 * sqrt14`  =`sqrt(5 * 14)`
                                                                = `sqrt70`
        Answer: `sqrt70` 
Examples for radical product rule math problem 2:
              Multiply the two math radical `sqrt 18 * sqrt 3`
        Solution:
                            Given radicals `sqrt18 * sqrt3`           
               We know the math radical product rule  `sqrt(ab)`   =  ` sqrta * sqrt b` 
                                     So,  `sqrt18 * sqrt3` =`sqrt(18 * 3)`
                                                                = `sqrt54`
        Answer:  `sqrt54`
Math Division rules for radical:
                Square root division rule:        ` sqrt(a/b)` = `sqrta / sqrt b`
               Cube root division rule:          ` root3 (a/b)` =  `root3 (a) / root3 (b)`
               n^th root division rule                 `rootn (a/b)` = `rootn (a) / rootn (b)`
Examples for radical division rule math problem 1:
              Simplify  the math radical ` root3 (66) / root3 (11)`
       Solution:
                            Given math radicals  ` root3 (66) / root3 (11)`      
               We know the math radical division rule    ` root3 (a/b)` =  `root3 (a) / root3 (b)`
                                                So, ` root3 (66) / root3 (11)`   = `root3 (66/11)`
                                                                      = `root3 6`
        Answer: `root3 6`
Examples for radical division rule math problem 2:
              Simplify  the math radical ` root4 (16) / root4 (24)`
       Solution:
                            Given radicals ` root4 (16) / root4 (24)`      
               We know the math radical division rule    `rootn (a/b)` = `rootn (a) / rootn (b)`
                                              So, ` root4 (16) / root4 (24)`   = `root4 (16/24)`
                                                                      = `root4 (2/3)`
        Answer:  `root4 (2/3)`

Other Radical rules in math:

Relation rules with exponential term:
                      `root(n)(x)`^m =( `root(n)(x)` )^m = (x^1/n )^m = x^m/n
Examples for radical with exponent math problem 1:
              Simplify the math radical `"(root5 4)^5 * `
        Solution:
                     Given math radicals `(root5 4)^5 * (sqrt27)^2`
                                                = `(root5 4)^5 * (sqrt27)^2`
              we know the math radical rule with an exponents  ( `root(n)(x)` )^m = x^m/n
                          So,     = 4^5/5 * 27^2/2
                                  = 4¹ * 27¹
                                              = 4 * 27
                                              = 108
        Answer:   108
Examples for radical with exponent math problem 2:
              Simplify the math radical `(root4 5)^3 * (sqrt5)^3`
        Solution:
                     Given radicals `(root4 5)^3 * (sqrt5)^3`
                                              = `(root4 5)^3 * (sqrt5)^3`
              we know the math radical rule with an exponents  ( `root(n)(x)` )^m = x^m/n
                          So,     = 5^3/4 * 5^3/2
                                  = `5` ^{(`3/4` + `3/2` )}
                                              = 5^9/4
             Answer:  5^9/4

Calculus Help with Assignment

Calculus (Latin, calculus, a small stone used for counting) is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.

(Source: Wikipedia)

Example problems for calculus help with assignment

Calculus help with assignment problem 1:

Differentiate the given function with respect y and find the value of g'(y). The given function is g(y) = 0.45y3 + 3.3t2 - 4y4 + y5

Solution:

The given function is g(y) = 0.45y3 + 3.3t2 - 4y4 + y5

Differentiate the given function with respect to x, we get

g'(y) = (0.45 * 3)y2 + (3.3 * 2)y - (4 * 4)y3 + 4y4

= 4y4 - 16y3 + 1.35y2 - 6.6y

Answer:

The final answer is 4y4 - 16y3 + 1.35y2 - 6.6y

Calculus help with assignment problem 2:

Differentiate the given equation with respect to t, y = 23t5 + 42t3 - 9t2 - 16

Sol:

Given y = 23t5 + 42t3 - 9t2 - 16

Differentiating both sides, we have

`(dy / dt)` = (23 * 5)t4 + (42 * 3)t2 - (9 * 2)t - 0

= 115t4 + 126t2 - 18t

`(dy / dt)` = 115t4 + 126t2 - 18t

Answer:

The final answer is 115t4 + 126t2 - 18t

Calculus help with assignment problem 3:

Integrate the given function ∫ (9x2) dx

Solution:

The given function is 9x2 dx

Integrate the given function with respect to x, we get

∫ (9x2) dx = 9 `(x^3 / 3)` + c

= 3x3 + c.

Answer:

The final answer is 3x3 + c

Calculus help with assignment problem 4:

Integrate the given function g(x) and find the value of G(3). The given function is g(x) = 14x6 - 5x4 + 28x + 40

Solution:

The given function is g(x) = 14x6 - 5x4 + 28x + 40

Integrate the given function with respect to x, we get

G(x) = 14 `(x^7 / 7)` - 5 `(x^5 / 5)` + 28 `(x^2 / 2) ` + 40x + c

= 2x7 - x5 + 14x2 + 40x + c

Find the value of G(3):

Substitute x = 3 in the above equation, we get

G(3) = 2 (3)7 - (3)5 + 14 (3)2 + (40 * 3)

= 4374 - 243 + 126 + 120

= 4377

Answer:

The final answer is 4377

Practice problems for calculus help with assignment

Calculus help with assignment problem 1:

Find the value f'(2). Given function f(x) = 12x3 - 4x2 + 9

Answer:

The final answer is 128

Calculus help with assignment problem 2:

Integrate the given function g(x) = 6x2 + 12. Find the value G(4).

Answer:

The final answer is 176

Percent Composition

Per cent composition of the element in the molecule is defined the ratio of the mass of the different elements present in the molecule expressed in term of percentage.    The easiest method for  determining the percentage mass of a particular element in the compound is to find out the molecular mass of the element.  The molecular mass of the element is calculated by adding all the atomic mass of the element in a compound  The find out the mass of the particular element that is represented in the molecule

The next step is to divide the mass of element upon the molar mass of the molecule and converting them in percentage.   For example for determining the % composition of the A in A3B4

% of A       =    (3 x atomic mass of A/(3 x atomic mass of A + 4 x atomic mass of B)) x 100

Example 1 for the determination of percent composition

Q. Calculate the % composition of  each element in NaOH?

Ans:  There are three element in NaOH.  Sodium, oxygen and hydrogen.

The molecular mass of NaOH = 23 + 16 +1 = 40g/mole

% composition of Na = (23/40) x 100 =57.5%

% composition of Oxygen = (16/40) x 100 = 40%

% composition of Hydrogen = (1/40)  x 100 =  2.5%

Example of percentage composition 2

Q.  Calculate the percentage composition of H2SO4?

Answer:  There are three element in H2SO4 they are hydrogen, sulfur and oxygen

The molecular mass of H2SO4 = 2x 1+32 + 4 x16 =98g/mole

% composition of Hydrogen = (2/98) x 100 = 2.04%

% composition of sulfur  = (32/98) x 100 = 32.65%

% composition of oxygen   = (64/98) x 100 = 65.31%

Excercise

Find the % composition of Carbon in carbon dioxide?
Find the % composition of chromium in potassium dichromate
Find the pecentage composition of water in copper(II) sulfate pentahydrate
Find the percentage composition of Mn in KMnO4

Expression Design Tutorial

An expression design tutorial is a finite combination of symbols that are well-formed according to the rules applicable in the context at hand. Symbols can designate values constants, variables, operations, relations, or can constitute punctuation or other syntactic entities. In algebra an expression may be used to designate a value, which value might depend on values assigned to variables occurring in the expression. Mathematical expressions include letters called variables.

(Source -Wikipedia)


Basic concepts of Expression design tutorial:

Basic concepts of Expression design tutorial:

Basic concepts of Expression:

Expression design Tutorial mean nothing but a to make the expression using constant, Operations, Variables, according to our given sign and operation of the expression.

Example:

A simple algebraic expressions like  6x + 8, 7y – 9. A variable can take various values. Its value is not fixed. On the other hand, a constant has a fixed value. Examples of constants are: 4, 100, and 17.

Based on the number of of terms expression should classified four types:

Monomials(having only one term)
Binomials(having two terms)
Trinomials(having three terms)
Polynomials(having many terms)

In word problems data’s are given in directed form when we solve the word problem first design the expression based on our given data and then solve the expression

Example problems in expression design tutorial:

Expression design tutorial:

Example problem 1:

16 years ago, Jenny age was half of the age his brother will be in 25 year.find the current age of jenny?

Here first we have to design the expression for solving jenny’s age

Solution:

Step 1:

Let us consider x is age of Jenny

14 years ago means that x-16

Step 2:

Half of the age of his brother will in 25years means =x+25

Step 3:

Now we have to write the equation or expression for solving age

x-16=1/2(x+25)(Designed expression)

Step 4:

2(x-16) =x+25(solving the expression)

2x-32=x+25

Step 5:

Subtract both side on x

2x-32-x=x-x+25

2x-x-32=25


Step 6:

Add both sides on 32

2x-x-32+32=25+32

Step 7:

X=25+32

X=57 year

Answer :jenny age was 57 year

Tangent Trigonometry

The word trigonometry is an origin of two Greek words “trigonon” meaning a triangle and “metron” meaning measurement. Metron means the science, which deals with the measurement of triangles. The familiar trigonometric terms are sine, cosine, tangent, cotangent, secant, cosecant. The tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space
                                                                                                                                                                                    - Source from Wikipedia

Looking out for more help on Trigonometry Formulas Sheet in algebra by visiting listed websites.

Definition of tangent trigonometry:

                A function of an acute angle in a right-angled triangle can be defined as the ratio of the opposite side of length and the adjacent side of the length is called tangent. It is expressed by the gradient of a line. Find either sides or angles in a right-angled triangle by using this function.
      
  In Diagram (A) represents the Thangent to curve
 In Diagram (B) represents the tangent of an angle
 In Diagram (c) represents the tangent to a circle.
                       In Right angle triangle Tangent  can be expressed in mathe matical form. If an angle is` theta`
               ` tan theta ` = `(Opposite side)/(adjacent side)`

Tangent trigonometry problems:

Tangent trigonometry problem 1:
          Find the height of the given right-angle triangle.
           
         Solution:
                               tan 30° = `(opposite)/(adjacent)`
                                             = `h/5`                                                                                 we know, tan 30°= `1/ sqrt3 `

                       height  (h) = 5 × tan 30°
                                           = 5 × `1/ sqrt3 `
                                          =  2.886 (approximately)
      So, The height of the right-angle triangle (h) =2.886 cm


Tangent trigonometry problem 2:
           Find the hypotenuse value of the given right-angle triangle. length and height are in centimeter.

                
      Solution:
                                           Tan` theta`   = `(opposite side)/(adjacent side)`
                                                       = `4/3`
                                   Tan `theta` = 1.333
                                            `theta` = tan^-11.333
                                            theta = 53.12°
                  Now the trigonometry relation,  sin` theta` = `(opposite side) / (hypotenuse)`
                                                               sin 53.12° =` 4 / (hypotenuse)`
                                                     hypotenuse (X) =` 4 / sin 53.12 `                                we know sin 53.12° = 0.7999
                                                     hypotenuse (X) = `4 / 0.7999`
                                                     hypotenuse (X) = 5.000
                  So, the hypotenuse value of right-angle triangle is 5 cm.

Descriptive Geometry

The descriptive geometry is a branch of geometry, which deals three dimensional objects in two dimensions by using the lines, curves, solids, surfaces and points in space. Geo means “earth” and metron  means “measurement”. ”Euclid, is a Greek mathematician, called the father of geometry. A point is used to represent a position in space. A plane to be a surface extending infinitely in every directions such that all points lying on the line joining any two points on the surface. The descriptive geometry example problems and practice problems are given below. 

Example problems for descriptive geometry:

Example problem 1:
          Show that the straight lines 2x + y − 9 = 0 and 2x + y − 10 = 0 are parallel.
Solution:
           Slope of the straight line 2x + y − 9 = 0 is m₁ = − 2
           Slope of the straight line 2x + y − 10 = 0 is m₂ = − 2 ∴ m₁ = m₂
           The given straight lines are parallel.

Example problem 2:
      Find the co-ordinates of orthocentre of the triangle formed by the straight lines x − y − 5 = 0, 2x − y − 8 = 0 and 3x − y − 9 = 0
Solution:
      Let the equations of sides AB, BC and CA of a ΔABC be represented by
                x − y − 5 = 0 … (1)
               2x − y − 8 = 0 … (2)
               3x − y − 9 = 0 … (3)
      Solving (1) and (3), we get A as (2, − 3)
       The equation of the straight line BC is 2x − y − 8 = 0. The straight line perpendicular to it is of the form
             x + 2y + k = 0
       A(2, − 3) satisfies the equation (4) ∴ 2 − 6 + k = 0 ⇒ k = 4
       The equation of AD is x + 2y = − 4 … (5)
       Solving the equations (1) and (2), we get B as (3, − 2)
       The straight line perpendicular to 3x − y − 9 = 0 is of the form x + 3y + k = 0
       But B(3, − 2) lies on this straight line ∴ 3 − 6 + k = 0 ⇒ k = 3
       The equation of BE is x + 3y = − 3 … (6)
       Solving (5) and (6), we get the orthocentre O as (− 6, 1).

Example problem 3:
         Find the values of a and b if the equation (a − 4)x² + by² + (b − 3)xy + 4x + 4y − 1 = 0 represents a circle.
Solution:
   The given equation is (a − 4)x² + by² + (b − 3)xy + 4x + 4y − 1 = 0
            (i) coefficient of xy = 0 ⇒ b − 3 = 0 ∴ b = 3
            (ii) coefficient of x² = co-efficient of y² ⇒ a − 4 = b
                        a = 7
                Thus a = 7, b = 3

Practice problems for descriptive geometry:

Practice problem 1:
          If the equation 12x² − 10xy + 2y² + 14x − 5y + c = 0 represents a pair of straight lines, find the value of c. Find the separate equations of the straight lines and also the angle between them.
     Answer: C = 2 ; 6x − 2y + 1 = 0 and 2x − y + 2 = 0 ; tan⁻¹ (1/7)
Practice problem 2:
          Find the equation of the straight line which passes through the given intersection of the straight lines 2x + y = 8 and 3x − 2y + 7 = 0 and is parallel to the straight line 4x + y − 11 = 0
      Answer: 28x + 7y − 74 = 0