Simple Equations and Inequations

An equation

An equation is a condition of variable. The word variable means something whose value can be changed. The value of variable is not fixed. They are usually denoted by small alphabets like x, y, z, u, v, w etc. Expression can be formed using various operations like addition, subtraction, multiplication and division on variables.

In an equation, there is always a sign of equality. This equality sign indicates that the value of the expression to right side of equality sign should be equal to the value of the expression to the left side of equality sign.

(ii) At least one of the two expressions (LHS or RHS) of an equation should have variable involved.

(iii) An equation remains the same when the expression on the left and right side are interchanged.

An algebraic expression with sign of equality having one variable with highest power of variable as one e.g. 3x + 4 = 0 is an linear equation

Linear equation is an equation, in which variables have power of one only.

(i) Read problem carefully and find what is given and what is required

(ii) Identify the unknown quantity and denote it by any variable like x, y, z etc

(iii) Translate the statement of the problem into a mathematical sentence i.e., an equation

Linear Inequations

Linear Inequations: A statement of inequality between two expression involving a single variable with highest power 1, is called a linear inequation.

Example. Each of the statements

(i) x < 2

(ii) 3x + 1 = 4

(iii) x/3 > 5

(iv) 2x – 3 > 7

Domain of the Variable or Replacement Set

The set from which the values of the variable x are replaced in an inequation, is called the domain of the variable or the replacement.

Example: Consider the inequation x < 3 and let the replacement set be N. Here, we can replace x only by some member of N. Clearly, some values of x from N will satisfy the inequation x < 3 while some other values of x from N will satisfy it. Some other values of x from N will not satisfy it

The set of those values of x from N which satisfy the inequation x < 3, is the solution set of the given inequation.

Thus, we define the solution set of an inequation as under.

Solution Set

Solution Set: It is the subset of the replacement set, consisting of those values of the variable which

satisfy the given inequation.

Example. Find the solution set of each of the following inequation:

(i) x < 4, where replacement set is N

(ii) x > - 2, where replacement set is A = {- 2, - 1, 0, 2, 3}

(iii) x < - 3, where replacement set is

B = {- 7, - 6, - 5, - 4, - 3, - 2, - 1}

Solution. We have

(i) Solution set = {x ? N : x < 4} = {1, 2, 3}

(ii) Solution set = {x ? A : x < - 2} = {- 1, 0, 1, 2, 3}

(iii) Solution set = { x ? B : x < - 3} = {- 7, - 6, - 5, - 4}

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