 Hello and welcome to the session. In this session we are going to learn how to recognize inequalities and solutions of inequalities and we shall also learn how to represent solutions on the number line. How can we write the given statement mathematically? The number of chocolates in a packet are more than 20. Here let the number of chocolates be x. Now we are given more than 20. It means greater than 20. So we write the number of chocolates that is x is greater than 20. There are a few phrases that help us to recognize inequalities from statements. Now for symbols like less than greater than less than or equal to greater than or equal to we have a few phrases like for less than we have less than and is fewer than. Similarly for greater than we have greater than and is more than for less than or equal to we have is less than or equal to is at most is no more than and for greater than or equal to we have is greater than or equal to is at least is no less than. Now let's say Alice have at most 100 dollars in her pocket and out of these she wants to buy two dresses of 40 dollars each but she also wants to buy as many pairs of earrings that she can buy of 10 dollars each. Here how can Alice figure out how many pairs of earrings she can buy? For this we can form an inequality. Here we have Alice has at most 100 dollars in her pocket at most 100 dollars means Alice has 100 dollars or less in her pocket that is less than or equal to sign. With this Alice can set up an inequality to determine how many pairs of earrings she can buy. First she spends 40 dollars each on two dresses that is two dresses would cost her two into 40 dollars that is two into 40 which is equal to 80 dollars as she wants to find the number of pairs of earrings so she can take this unknown number as X so let the number of pairs of earrings be X then the inequality becomes 80 that is the cost for two dresses plus 10 into X as the cost of X number of pairs of earrings each of 10 dollars will be 10 X is less than or equal to 100 so we got the inequality as 80 plus 10 X is less than equal to 100. Now she can easily find the number of pairs of earrings now we are going to learn how we can solve an inequality by using addition, subtraction, multiplication and division. Suppose that X plus 5 is greater than 10 is an inequality that we want to solve for X to solve this inequality for X we shall keep X on one side of the inequality in order to do this we shall subtract 5 from both the sides of the inequality we have this inequality that is X plus 5 is greater than 10 when we subtract 5 from both the sides we get X is greater than 10 minus 5 that is 5 so we get X greater than 5 the solution of this inequality this was a simple inequality that could be solved in one step. Now if we consider the inequality found in the example above we have 80 plus 10 X is less than equal to 100 here X is the number of pairs of earrings that Alice wants to buy so in this inequality we want to find the value of X so we try to keep X alone on one side of the inequality so we subtract 80 from both the sides and we get 10 X is less than equal to 20. Now we divide both sides by 10 as on dividing both sides by a positive number will not change the sign of the inequality and therefore we get 10 X by 10 is less than equal to 20 upon 10 which implies that X is less than or equal to 2 so we can conclude that here X that is the number of pairs of earrings at most 2 the inequality was 80 plus is less than equal to 100 and the solution to this inequality is given by X is less than or equal to 2 so we can say this inequality has a two step solution obtained by using addition and then division now let us plot the solution on the number line first we draw a number line with positive numbers to the right of 0 and negative numbers to the left of 0 now here is the number line with positive integers to the right of 0 and negative integers to the left of 0 as in the above example we have the inequality that is X is less than equal to 2 so we mark 2 on the number line with a closed dot as X is less than or equal to 2 and 2 is included in the solution and then we shape the portion to the left of 2 as all numbers to the left of 2 will be less than 2 thus from the number line we can see that an inequality has more than one solution here all numbers less than or equal to 2 its solution if the inequality was X less than 2 then 2 would be marked with an open dot as 2 would not have been included then and in this way X is less than 2 will be shown on the number line let us take one more example suppose we want to solve this equation that is 2X plus 5 is greater than 3 for X for this we first subtract 5 from both the sides that is we have the equation 2X plus 5 is greater than 3 and we subtract 5 from both the sides so we get 2X is greater than minus 2 as 3 minus 5 will be equal to minus of 2 now we divide by 2 on both the sides and we get 2X upon 2 is greater than minus 2 upon 2 and therefore we get X is greater than minus 1 and this is the solution now let us plot it on the number line and here is the number line in which positive integers are to the right of 0 and negative integers are to the left of 0 now we have to plot X greater than minus 1 on this number line here we mark minus 1 with an open dot on the number line as minus 1 is not included now we set the portion on the right of minus 1 as X is greater than minus 1 in which all the numbers greater than minus 1 are included we should note that when we multiply or divide an inequality by a negative quantity then finally inequality will get reversed here we take an example if we have to solve the equation that is 3 minus 2X is greater than 5 for this first we subtract 3 from both the sides so we have the equation that is 3 minus 2X is greater than 5 and if we subtract 3 from both the sides we get minus of 2X is greater than 5 minus 3 that is 2 now we can solve this inequality by the following method first add 2X to both the sides and therefore we get minus 2X plus 2X is greater than 2 plus 2X which is equal to 0 is greater than 2 plus 2X now we subtract 2 from both the sides to make X alone on one side of the inequality and therefore we get 0 minus 2 is greater than 2 plus 2X minus 2 which is equal to 0 minus 2 is minus 2 is greater than 2X and now we divide both the sides by 2 and we get minus 2 by 2 is greater than 2X by 2 which is equal to minus 1 is greater than X and minus 1 is greater than X is the required solution of the inequality if we consider the above inequality again and if we divide both the sides by minus 2 we get minus 2X by minus 2 is greater than 2 by minus 2 which is equal to is greater than minus 1 in our previous result we have called that X is less than minus 1 and here we get X is greater than minus 1 so we can say this is not the right method to solve inequality and our answer is wrong and if we compare this answer with our previous answer which is correct then we can clearly see that if we multiply or divide an inequality by any negative quantity then sign of the inequality will get reversed let us take one more example in this we know that 5 is greater than 2 but we also know that minus 5 is less than minus 2 so we can clearly see that if we multiply an inequality by any negative quantity that is minus 1 in this example then sign of the inequality will get reversed this completes our session hope you enjoyed this session