=> 2x + 5 - 5 = 13 - 5

=> 2x = 8

=> x = 8/2 = 4

Hence We can see the solution being represented on the number line.

A statement of inequality between two expressions involving a single variable x with highest power 1, is called a linear inequality..

variable.

A = { 1,2,3,4,5,6,7,8}

**Solution:** We have B = { } , since there is no positive integer in this interval.

When 6 > 4, then 6/2 > 4/2 => 3 > 2

=> - 12 < -8

When 6 > 4, then 6/(-2) < 4 / ( -2)

=> -3 < -2

Example : When 1/4 < 1/2, then 4/1 > 2/1

=> 4 > 2

2x + 5 > = 13

=> 2x + 5 - 5 > = 13 - 5

=> 2x > = 8

=> x > = 8/2

=> x > = 4

In the above number line, the darkened portion x > = 4, is the solution set for the inequality.

Here the variables used are x and y. As the value of changes the value of x also changes the value of y also changes correspondingly.

Since x can take infinitely many values on the real number line, there will be corresponding real value for y.

Example : Find any two solutions of the linear equation of two variables 4x + y = 8 hence represent it graphically.

When 4x + y = 8 => y = 8 - 4x

Let us assume some values for x

when x = -2, y = 8 - 4 ( -2) = 8 + 8 = 16

Hence

When x = 0, y = 8 - 4 ( 0 ) = 8 - 0 = 8

The

Hence by substituting different values for x, we can find the corresponding values of y.

The pair of values (x,y) are called the set of solutions to the given equation.

When we plot the points on the two-dimensional graph, we get a straight line. The line will divide the graph into two regions.

The graph will be as shown below.

The general form of linear inequalities of 2 variables will be of the form ax + by < c, ax + by < = c, ax + by > c, ax + by > = c.

We know that the graph of the equation ax + by = c, is a straight line which divides the xy-plane into two parts which are represented by ax + by >=c,

and ax + by < = c. These two parts are known as closed half spaces.

The region ax + by < c and ax + by > c are known as the open half spaces.

These half spaces are known as the solution sets of the corresponding inequalities.

In order to find the solution set of a linear equation in two variables, we follow the following steps.

Example : Represent the region which satisfy the two inequalities 4x + y <= 8 , and 15 x + 7 y > 105 in two dimensional graph.

Let us find the x and y intercepts of each line.

4x + y = 8

x |
y |

0 |
8 |

2 |
0 |

(1) We draw thick line in the graph.

(2) when we plug in (0,0) , we get 4(0) + 0 < 8 => 0 < 8, which is a true statement.

Hence we shade the region which does contain the origin.

15 x + 7y = 105

x |
y |

0 |
15 |

7 |
0 |

Since the inequality is 15 x + 7y > 105,

(1) We draw the dotted line in the graph.

(2) By substituting (0,0) we get 15(0)+7(0)> 105 => 0 > 105,

which is a false statement.

Hence we shade the region which does not contain the origin. The graphs of these two inequalities are shown below.

As the value of x varies, the value of y also varies.

The slope - intercept form of the linear equation of two variables, y = mx + b, where 'm' is the slope and 'b' the y-intercept of the line.

The graph of the line is a straight line which can be extended indefinitely on both the direction.

The Domain of the function is the set of all real numbers,

x = { x : x belongs to Real numbers }

The range of the function is also the set of all real numbers which depen on x.

y = { y : y belongs to real numbers, y = f(x) }

Find the Domain and Range of the function, where 0 < x < 5 and x belongs to Natural numbers

Let us write it in the form y = f(x).

2x + y = 6

=> y = -2x + 6

The domain likes in the interval (0,5), where x belongs to Natural numbers.

Therefore,

To find the Range:

Substituting x = 1, we get, y = -2(1)+6 = -2+6 = 4

Substituting x = 2, we get, y = -2(2)+6 = -4 + 6 = 2

Substituting x = 3, we get, y = -2 (3) + 6 = -6+ 6 = 0

Substituting x = 4, we get, y = -2(4) + 6 = -8 + 6 = -2

Therefore

2. Check if x = 4 is the solution of the equation 6x + 4 = 20

3. Find any four solutions of the equation 3x + y = 10.

4. Find the solution set of the inequality, 2x + 5 < = 15, where x belongs to Real numbers.

Express your answer in interval form and also represent on the real number line.

5. Graph the equation 4x + 7y = 28, by finding the x and y intercepts.

6. Shade the region which satisfy the inequality 3x + y > = - 2.

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