Solve absolute value inequalities

Introduction:

        Other than algebra, there is one more important inequality. It is called as Absolute value inequality. The absolute value for any number is numerical value regardless of its sign.  For example, the absolute value of | -5 | is 5 and │+5 │ is 5. Here the vertical lines denotes absolute value.

It is a little more difficult when dealing with equations. There are three possible outcomes for absolute value inequalities. 

1. If x is any expression and a any positive number, and │x │= a, then it has either x = a or x = - a values.
2. If x is any expression and a any positive number, and │x │<>
3. If x is any expression and a any positive number, and │x │> a then it looks   x < -a and x > a.

Example

Solve: │4x – 4 │ = 8

Solution:

Use the result, if |x| = a, then x = a or x = -a

Here │4x – 4 │ = 8

So, it has either 4x – 4 = 8 or  4x – 4 = -8 values.

Solve each equation using the addition and multiplication principles.

4x = 12,    4x = -4

after solving the equation, we get

x = 3,     x = -1. Answer

Hope you like the above explanation to solve the absolute value inequalities, Please leave your comments, if you have any doubts.

Practice of Algebra Application

Introduction:

 Applications of algebra are found everywhere; the principles of algebra are applied in all branches of mathematics, for instance, calculus, geometry, and topology. They are applied every day by men and women working in all type of business. As a typical example of applying algebraic methods.

     Algebra is often referred to as a generalization of arithmetic, so it is collection of rules, then the rules for translating words into symbolic notation of mathematics, rules for formulating statements using symbolic notation, and rules for rewriting mathematical statements in  a manner that leaves their truth unchanged. 

Application of Algebra in Slopes of Straight Lines:

In general slope of equation is y=mx+b
Practice  Problem 1:
Y=x-3
Where m=1 (co efficient of x)
               B=-3
Equation of point-slope :
 Formula for point,slope
                (y-y1)=m(x-x1)

Practice  problem 1.  Find the equation of the line which passes through the point (2, 2) and has slope3.
Solution:
       Equation is    (y-y1)=m(x-x1)
Where, x1=2
                Y1=2
                M=3
So,
        y-2=3(x-2)
        y-2=3x-6
Answer is: 3x-y-4=0

Learn factor polynomials calculator

Introduction:

         Calculator is used to solve different types of problems. It is a web-based tool designed to solve different problems. Learning factor polynomials through calculators is simple. 

      Factor polynomials calculator are used to understand the polynomials factorization. Given expression can be factorized by using the greatest common factor. Let us discuss about the learn factor polynomials calculator.

Steps to learn factor polynomials:

The Steps to learn factor polynomials are as follows:

• Given expression can be arranged in the order of powers.
• Expression can be in the form of standard ax² + bx + c = 0.
• The expression should be factorized.
• Solve the given terms.

Example problems to learn factor polynomials calculator:

Factorize: 42ax + 36bx – 7ax²– 6by² using factor by group
Solution:
Step 1:
          Given expression 42ax + 36bx – 7ax²– 6by²
Step 2:
           Given expression in the standard form ax² + bx + c = 0
                     42ax + 36by – 7ax²– 6by² = 0
Step 3:
           Groups the terms
                      42ax – 7ax² +36by – 6by² = 0
                      7ax (6 – x) + 6by (6 – y) = 0
Step 4:
           The greatest common factor for the expression
                      7ax (6 – x) + 6by (6 – y) = 0
                     (7ax + 6by) (6 – x) (6 – y) = 0
Solution to the given equation is (5a – 6x) (3b – 4y) (b – 3y) = 0.

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About Polynomials

Introduction:

           An algebraic expression in which the variables have only non negative powers is called polynomial.  The monomial midterm study which contains finite number sum in x it said to be polynomial. The coefficients of the polynomial are said to be coefficients of the monomial in a polynomial. If coefficients of a polynomial are zero, then polynomial is said to be  zero polynomial. The highest coefficient power of x in a polynomial is called the leading coefficient for polynomial

Fundamental of Polynomial

Constants and variables are the fundamentals of polynomials.
Examples :In the formula for circumference of a circle, c=2π r ² and π are constants and c and r are variables

Degree and Types of Polynomials

Polynomials with more than one variable, the sum of powers of the variables taken up and the highest sum so obtained is called the degree of the polynomial.
Monomial, Binomial and Trinomials are three different types of Polynomials.

Properties of Polynomials

Addition and subtraction of two polynomials mean combining like terms.
We can perform multiplication and division also

Factorization of Polynomials

     You know that any polynomial of the form p(a) can also be written as
p(a) = g(a) x h(a) + R(a) it implies that Dividend = Quotient X Divisor + Remainder.
If the remainder is zero, then p(a) = g(a) x h(a). That is, the polynomial p(a) is a product of two other polynomials g(a) and h(a).

Order of operation using algebra

Introduction of order of operation using algebra:
 The Order of Operation using algebra is
P          - Parentheses,
E          - Exponents,
M         - Multiplication,
D          - Division,
A          - Addition and
S          - Subtraction

1. Calculations have to be finished from left to right in algebra.
2. First complete the operations inside a parenthesis.
3. Next Complete the operations of  exponents.
4. Then do multiplication and division, from left to right.
5. Then  do addition and subtraction, from left to right.

Step by step process for order of operation:

Parenthesis

                In algebra the parenthesis, combining symbols are always completed from the innermost set outward.

Exponents

                In algebra, an exponent is unlike than just multiplying. 

Example: 2^4 means 2 x 2 x 2 x 2 = 16

Evaluate 2 + 5 x (4 + 3) ÷ 2 - 6 using the order of operations.

Solution: 

Step 1:	2 + 5 x (4 + 3) ÷ 5 - 6	=	2 + 2 x 9 ÷ 3 – 6	Parentheses
Step 2:	2 + 5 x 7 ÷ 5 – 6	=	2 + 18 ÷ 3 – 6	Multiplication
Step 3:	2 + 35 ÷ 5 – 6	=	2 + 6 – 6	Division
Step 4:	2 + 7 - 7	=	2+6 – 6	Addition
Step 5:	2	=	2	Subtraction

Multiplication and Division in algebra

                These operations are completed in the order they show from left to right.  They are finished together because they have same importance.

Addition and Subtraction in algebra

         These two operation have two operations of the same importance. These are finished in the order that they show from left to right. 

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Learn Greatest Common Factor

Introduction:
        Greatest Common Factor (GCF) is the greatest number that is a factor of each of two or more numbers. To find either the Least Common Multiple (LCM) or Greatest Common Factor (GCF) of two numbers, you always start out the same way: you find the prime factorizations of the two numbers. Then you put the factors into a nice neat grid of rows and columns, and compare and contrast and take what you need.
Let us study the GCF of two numbers 24 and 15.
The factors of 24 are 1,2,3,4,6,8,12 and 24.
The factors of 15 are 1,3,5 and 15.
From this we observe that the greatest factor common to both these numbers are 3. In other words we can say that the greatest common factor of 24 and 15 is 3.

Greatest Common Factor Learning

Another method of finding out  the GCF is by using prime factorisation. The prime factors of 8 are 1,2,2,2 and that of 12 are 1,2,2,3. The prime factors that is  common for both numbers are 1,2,2. Multiply together all these numbers and we get 4. We can say that 4 is the greatest common factor of these numbers.

Least Common Multiple

Introduction:
   In arithmetic number of theory is the least common multiple or lowest common multiple (LCM) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both of a and of b. Since it was a multiple, it can be divided by a and b without a reminder. If either a or b is 0, so that number is no such positive integer, then LCM(a, b) is defined to be zero.

Example:

Multiples of 4 are
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, ......................
Multiple of 7 are
7, 14, 21, ................
(add 4 to each get to next multiple)

Applications

When adding and subtracting, or comparing vulgar fractions, Useful to find the least common multiple of the denominators, is  called the lowest common denominator. For instance,
{3\over15}+{1\over4}={12\over60}+{15\over60}={9\over20},
where the denominator of 60 was used to least common multiple of 15 and 4.

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How to simplify Negative Exponents

Introduction:
      Exponents are also called Powers or Indices. Exponents containing negative sign are called negative exponents. The exponent of a number  says how many times to use the number in a multiplication. Dividing is the inverse (opposite) of Multiplying. A negative exponent number says how many times to divide by the number.
     Calculate the positive exponent (aⁿ)
       Then take the Reciprocal (i.e. 1/aⁿ)
      To change the sign ( minus to plus or plus to minus) of the exponent, use the Reciprocal (i.e. 1/aⁿ).

Simplifying Negative Exponents

Simplifying Negative Exponents:

            Exponent  is a small number written near the top of another number which show how many times a number or a variable is multiplied by itself. Negative exponents  indicate the inverse of the corresponding positive exponent.

Simplifying Variables with Negative Exponents:

Consider a problem like the one shown below.
x^-4  = 1/x⁴

Simplifying Fractions with Negative Exponents:

To increase a fraction to a power, raise the numerator and denominator to that power.
            a^-3b-⁴/b^-2=a-³b^-2


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Arithmetic Problems

Definition:

Arithmetic involves the study of quantity, especially as the result of combining numbers. In common usage, Arithmetic refers to the simpler properties when using the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers.

1) Question: The arithmetic mean between two terms in an arithmetic sequence is 21. If one term is 13, then find the other term.

A ) 58
B ) 50
C ) 29
D ) 21

Steps to derive

1 (a+b)2
[The arithmetic mean of two numbers a and b.]

2 (13+b)2 = 21
[Substitute the values.]

3 13 + b = 42

4 b = 29

5 So, the second term is 29.



Hence the right answer is Option C

2) Question: If 6 arithmetic means are introduced between 1 and 15, then find their sum.

A ) 64
B ) 48
C ) 2
D ) 32

Steps to derive

1 Let x₁, x₂, x₃, x₄, x₅, x₆ be the six arithmetic means introduced between 1 and 15 respectively.

2 1, x₁, x₂, x₃, x₄, x₅, x₆, 15 are in arithmetic progression.

3 Sum of the 8 terms = 8 / 2[1 + 15]
[The sum of the n terms S_n = n2[a₁ + a_n] where n is the number of terms, a₁ is the first term and a_n is the last term.]

4 = 4 × 16 = 64

5 Therefore, the sum of the six arithmetic means = x₁ + x₂ + x₃ + x₄ + x₅ + x₆ = 64 - 16 = 48



Hence the right answer is Option B

Linear Equations and linear Functions

Definition:

  

Linear Equations: A linear equation is defined as an algebraic equation in which every expression is either a stable (constant) or the product of a stable and a single variable. Linear equations can have one or more than one variables.  

  Linear functions:  A linear function is defined as a function whose chart (graph) consists of sectors of one straight line during its domain.

We can establish the linear function and it acquires value f (a) at ‘a’ and f (b) at ‘b’ by the following formula:
                                 
f(x)=f(a) (x-b)/(a-b) + f(b) (x-a)/(b-a)
   Because the first term is Zero (0) when x is ‘b’ and is f (a) when x is ‘a’, while the second term is Zero (0) when x is ‘a’ and is   f(b) when x is ‘b’.
   Finally we get,     y= mx+b


1. Example of Linear Equation:
Solve 3[2m - (7 - 3m)] = m - 21.           
Solution:
First we can solve the left hand side.
3[2m-(7-3m)]
Here first solving the inside term.
3(2m-7+3m) and then,
3(5m-7)
15m -21
Now compare with right hand side.
15m- 21= m-21
15m-m= 21-21
14m= 0
Therefore m=0
Answer: m=0

2. Example of Linear Function:
Solve 3(2m - 1) = 4(m + 5).
Solution:
3(2m - 1) = 4(m + 5)
6m-3=4m+20
6m-4m= 20+3
2m= 23
  m =23/2
  m =11.5
Answer: m=11.5