 Hello and welcome to the session. In this session we will discuss about inequalities. Two real numbers or two algebraic expressions related by the symbol this, this, this or this form and inequality. An inequality of the form AX plus B less than 0 is a strict inequality. An inequality of the form AX plus B greater than equal to 0 is a linear inequality in one variable X when we have A is not equal to 0 and inequality of the form AX plus BY greater than equal to C is a linear inequality in two variables X and Y when we have A is not equal to 0 and B is not equal to 0. Next we discuss algebraic solutions of linear inequalities in one variable and the graphical representation. Any solution of an inequality in one variable is a value of the variable which makes it a true statement. Then the set of the values of the variable which makes an inequality true statement is called the solution set of the inequality. Then we follow some rules to solve linear in equations like the first rule is equal numbers may be added to or subtracted from both sides and inequality without affecting the sign of inequality. Then next rule is both sides of an inequality can be multiplied or divided the same positive number when both sides are multiplied or divided by a negative number then the sign of inequality is reversed. Times of the inequality 4X plus 3 less than 5X plus 7. First we subtract 5X from both the sides so we have 4X plus 3 minus 5X is less than 5X plus 7 minus 5X and thus we get minus X plus 3 is less than 7. Now we subtract 3 from both the sides so we have minus X plus 3 minus 3 is less than 7 minus 3 which further gives us minus X is less than 4. Now we multiply both sides of this inequality by negative 1 so we have negative 1 multiplied by minus X and 4 multiplied by negative 1. Now we know that when we multiply both sides of an inequality by a negative number then the sign of the inequality is reversed so this sign changes to this and thus we get X is greater than minus 4. This is the solution of the given inequality and the solution set is given by minus 4 to infinity. If we to represent X less than a or X greater than a on the number line then what we do is we put a circle on the number a and dark line to the left or right of the number a and to represent X less than equal to a or X greater than equal to a on the number line we put a dark circle on the number a and dark the line to the left right of the number a. Now the inequality that we had considered in the above example that is this inequality we find that the solution for this inequality is X greater than minus 4. Now let's represent X greater than minus 4 on a number line. Now when we have X greater than minus 4 so in this case a would be minus 4 so we put a circle on minus 4 and we dark the line to the right of minus 4 that is this line. Now this is the solution X greater than minus 4. Now we discuss graphical solution of linear inequalities in two variables. We know that a line divides the Cartesian plane into two parts and each part is known as a half plane like a vertical line will divide the plane in the left and right half planes. Now this would be the left half plane and this would be the right half plane and a non vertical line will divide the plane into lower and upper half planes like this is the upper half plane and this is the lower half plane and a point in the Cartesian plane will either lie on the line or will lie in either of the half planes. Now we have the region containing all the solutions of an inequality is called the solution region. If we have that an inequality is of the type Ax plus By greater than equal to c or Ax plus By less than equal to c then the points on the line Ax plus By equal to c are also included in the solution region and if an inequality is of the form Ax plus By greater than c or Ax plus By less than c then the points on the line Ax plus By equal to c are not to be included in the solution region. Let's try and solve the inequality 2x minus y greater than equal to 1 graphically. For this first we will draw the graph for the line 2x minus y equal to 1. Now this is the line 2x minus y equal to 1 this line divides the plane into two half planes 1 and 2. Now what we do is we select a point say origin which is not on the line but which lies in one of the half planes like the origin lies in the plane 1 and we see if this point satisfies the given inequality like the given inequality is 2x minus y greater than equal to 1. Now when we put x equal to 0 and y equal to 0 in this we get this that is we have 0 is greater than equal to 1 which is false hence the plane 1 which contains the origin is not the solution region of the given inequality. This shaded portion is the solution region of the given inequality including the points on the line 2x minus y equal to 1. This is how we solve a given inequality graphically. Next we discuss solution of system of linear inequalities in two levels. Consider the system of linear inequalities x greater than equal to 3 y greater than equal to 2 let's try and solve this graphically. First let's consider the inequality x greater than equal to 3 let's find out the solution region for this inequality for this first we will draw the graph for x equal to 3 this is the line x equal to 3 now let's consider the origin 0 0 and let's see if this satisfies the given inequality x greater than equal to 3 when we put x equal to 0 in this inequality we get 0 is greater than equal to 3 which is false hence the region containing the origin is not the solution region for the inequality x greater than equal to 3 instead the region which does not contain the origin is the solution region now this shaded portion is the solution region for the inequality x greater than equal to 3 including the points on this line x equal to 3 next we consider y greater than equal to 2 for this we will draw the graph for y equal to 2 this is the line y equal to 2 now again we will consider the origin and let's see if the origin satisfies the given inequality y greater than equal to 2 when we put 0 in this inequality we get 0 greater than equal to 2 which is again false so the region containing the origin is not the solution region for the given inequality but instead the region which does not contain the origin is the solution region this portion shaded in green is the solution region for the inequality y greater than equal to 2 including the points on the line y equal to 2 now this region is common to both the solution regions so this shaded portion is the solution region for the system of linear inequalities this is how we solve the system of linear inequalities in two variables graphically this completes this session hope you have understood the concept of linear inequalities