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Analyzing Equations and Inequalities

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While analyzing equations and inequalities we focus on some points. For analyzing equations and inequalities we follow some points:
1. First solve the expression.
2. Then understand and use the Properties of Real Numbers.
3. Then simplify the equation and inequalities.
4. At last find the absolute values equation and inequality.
If equal sign is present in the equations, and equation is used to find the unknown variables. One method for solving the equation is substitution method.
Suppose we have an equation p + q = 2p – 1;
We know that equation is used to find the unknown variable.
Here in this equation we solve for unknown variable ‘p’;
On solving the equation for unknown variable ‘p’ we get:
⇒ p + q = 2p – 1;
⇒p = q + 1;
Now we use substitution method:
Now substitute the value of ‘p’ in the equation:
⇒ p + q = 2p – 1;
Put p = q + 1;
⇒ (q + 1) + q = 2(q + 1) – 1;
Now we solve the value of unknown variable ‘q’.
⇒q + 1 + q = 2q + 2 – 1;
Now combine like terms:
⇒2q – 2q = 1 – 1;
⇒0;
This is how we solve the equation by substitution method.
And in case of inequality, if we have two variables ‘m’ and ‘n’, then there are conditions for analyzing the inequality equations.
Suppose ‘m’ is less than equal to ‘n’;
⇒ m ≤ n;
Inequality present;
‘m’ is greater than equal to n then we can write:
⇒m ≥ n;
And if ‘m’ is greater than n then we can write;
⇒m > n;
And if ‘m’ is smaller than ‘n’ then we can write;
⇒m < n;
These all are the properties of inequality. If in any equation these symbols are present then we can say there is an inequality in the expression.

Expressions and Formulas

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An expression is defined as the Combination of two or more terms and can be written in the form of Relations between variables and constant values. The only difference between the expression and the equation is that the sign of equal to is missing in the expression, which exists in the equation.
Now, let us learn how to solve expressions and formulas. In order to solve the expressions and formulas, we will put the value of the variable‘s’ in the expression and thus get the numerical value of the expression. It will be clearer with the following example.
Let us consider the expression 2ab + 4c, where we have the value of ‘a’ as 2, b = 3 and c = 4.
Now we will place these numerical values in the given expression, so it will become:
2 * 2 * 3 + 4 * 4
= 12 + 16 = 28.
Sometimes the expressions are in form of powers of the variables too. In such case, we proceed as follows:
In case we have 2x^2 + 4y as an expression, where x = 2 and y = 3, then we will solve the expression as follows :
2 * 2^2 + 4 * 3
= 2 * 4 + 12
= 8 + 12 = 20
Now we will look at the formulas. Formulas are used to find the values if the variables are known. In case, if length and breadth of the Rectangle are known and we need to find the values of the perimeter of the rectangle and area of the rectangle, then for this we will apply the formula as follows:
Perimeter = 2 * (length + breadth)
Area = length * breadth.
This is called solving the values using the variables.
When we have to solve equations, we Mean that we are finding the value of the variable, which satisfies the equations. The equations can be Linear Equations of one variable or the pair of Linear Equations with Two Variables. Firstly we are solving equations with one variable.

In order to solve the equation, we need to use different methods. First method will be to get the value of the variable by hit and trial method. According to this method, we will find the value of the variable by trying different values for the variables in the equation. On solving if we get the left side of the equation equal to the right side of the equation, then we say that the particular value satisfies the equation. As we go on trying different values which satisfies the equations, so we call it hit and trial method.

Another method to solve the equation is by the Algebra method. According to algebra method, we say that the variables of the equation are taken to one side of the equation and the constants are taken to another side of the equation. Thus we say that on solving we get the value for the variable of the equation.

Now if we take the pair of linear equations with two variables, these equations can be solved either by the graphical method or by the method of algebra.
In case we solve the pair of equations and get the value of the two variables in the equation, we say that the two values of the variables will satisfy both the equations. If we draw the graph for the two linear equations, we will get the solution of the equation by finding the meeting Point of the equation. The coordinates of the meeting point is the solution to the given equation.
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Solving Absolute Value Equations

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By the term absolute value, we Mean that we are going to find the value of the equation, which is always a positive number. All absolute values have magnitude, but no direction and the absolute value is represented by | | sign, which we call as the modulus sign.

When we say the term solving absolute value equations, we mean that we are going to find the value of the variable in the form of magnitude only. Suppose we are given the equation as follows:
Y = | x|
Here we observe that whatever be the value of x, either a positive value or the negative value, the result will be a positive value, as the sign of modulus is placed with it. Let us take x as a positive number first, suppose 5. In such a situation, we say that |5| is a positive value. Thus the value of y = 5, for x = 5

Now let us take the value of x = -5, in case the value of x = -5, the value of y = | -5 |, y = 5
So we observe that either the value of x is positive or it is a negative Integer, we come to the conclusion that the value of y will be the modulus value of x, thus it will be a positive number.
So for each two values of x, we will get one value of y.
Still we have one situation, where for one value of x, we have one value of y and this value of x is 0. Since we know that zero either a positive number or a negative number will always be the same value, which is represented by the origin. The graph so plotted for this function will be in the form of English letter V, which passes through the origin.
The series of Real Numbers include the numbers which can be both rational and Irrational Numbers. These numbers are called real numbers as they are not imaginary. Let us look at the properties of real numbers one by one:

First we look at the closure property of real numbers, according to which we say that if we have two real numbers say a and b, then we say that the sum of two real numbers is also a real number. It means that the closure property holds true for the addition of real numbers.

Similarly, if we take the two real numbers ‘a’ and ‘b’, then the difference of two real numbers is also a real number. It means that the closure property holds true for the subtraction of the real numbers.
On the other hand, if we take the two real numbers ‘a’ and ‘b’, then the multiplication of two real numbers is also a real number. It means that the closure property holds true for the product of the real numbers. Similarly, if we take the two real numbers ‘a’ and ‘b’, then the division of two real numbers is also a real number. It means that the closure property holds true for the quotient of the real numbers.
Further looking at the properties of real numbers, we say that the associative and the Commutative Property of the addition and multiplication of real numbers holds true. But these two properties do not hold true for the subtraction and division of the real numbers. It means if we have the real numbers a, b and c, then by commutative property of real numbers, we have :
(a + b) = (b + a) and a * b = b * a
But a – b < > b – a and a/ b < > b / a
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Solving Inequalities Algebraically

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Talking into consideration Solving Inequalities algebraically such as : x + 3 > 0, was a simple and easy expression and we must remember that when the expression is multiplied by any of the negative number, then the sign of inequality changes.
To solve inequalities, we Mean finding all of its solutions. To solve the inequality, we need to find the value of the variable. So we say that a solution of an inequality is a number which when substituted for the variable, makes the existence of the inequality a true statement. Let us take the following inequality:
X – 2 > 5
Now if we substitute 8 for x, the inequality becomes 8 – 2 > 5. So we say that x = 8 is a solution of the inequality. On the other hand, we will substitute -2 for the value of x gives the false statement, which can be expressed as:
(-2) – 2 > 5.
So we say that x = -2 is not a true statement. So we say that x = -2 is not the solution to the inequality. We must remember that inequality always has many solutions. In order to solve the equations, there are certain manipulations of the inequality which does not change the solutions. We must remember certain rules for the solution of inequalities:
1. On adding or subtracting the same number on both sides, makes the inequality exactly same. So we say that the equation x – 3 > 5 is same as x – 3 + 3 > 5 + 3
Or x > 8 is the solution to the inequality.
2. In case we change the position of the left and the right side of the inequality, the sign of inequality changes, thus if we have:
X + 2 > 4,
Or 4 < x + 2 .

Solving Absolute Value Inequalities

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Absolute value can be defined as measure of how further a number is from 0. For example: ‘10’ is 10 away from zero and -50 is 50 away from zero. Absolute value of 0 is 0 and absolute value of 5 is 5. These all are examples of Absolute Value Function. Negative Numbers are not included in absolute value function. Suppose we have any negative number then it is necessary to remove negative sign from number. So we take only positive values and zero in case of absolute function. Absolute value function is represented by the symbol '|'. This symbol is known as bar. If we put any negative number between this symbol then we get positive number outside this symbol. Now we will see process of Solving Absolute Value Inequalities.
If (<,> <, >) these symbols are present in any expression then we can say that equation contains inequality in it. Let's see how to solve absolute value inequalities. Suppose we have | 2p + 3 | < 6, absolute value inequality.

Solution: Given inequality | 2p + 3 | < 6, now first we will solve linear inequality. So we can write given inequality as:

=> - 6 < 2p + 3 < 6, it means 2p + 3 is greater than -6 and less than 6. Now subtract 3 from inequality. On subtracting we get:

=> - 6 – 3 < 2p + 3 – 3 < 6 – 3, on further solving we get:

=> -9 < 2p < 3,

Now divide whole inequality by 2, on dividing we get:

=> - 9 / 2 < p < 3 / 2,

On solving | 2p + 3 | < 6 we get –9/ 2 < p < 3 / 2. This is how we solve absolute value inequalities.