7 th Grade 6 7 Solve Inequalities by Multiplication or Division

Chapter 6 Lesson 7 solve inequalities by multiplication or division you will be able to solve inequalities by using the multiplication or division properties of inequality the multiplication property and division property of inequality state that an inequality remains true when you multiply or divide each side of an inequality by positive number divide the following inequalities and graph the solution so again we're going to solve these just like we solve equations to get by itself we're going to divide both sides by 8 and we get less than or equal to 5 five goes in the middle of our number line followed by six and after four it is closed Circle and now need to pick number to plug in let's choose four plug it in where you see is four less than or equal to five that is true statement so four is our solution we shade or draw an arrow to the left letter / 2 is greater than 7 we need to multiply both sides by two we get oops is greater than 14 14 is our middle number then 15 and 13 before it it is an open circle and let's plug in number is 13 greater than 14 it is not this is false so let's try our other number is 15 greater than 14 that is true so we draw an arrow through the 15 letter divide both sides by 4 less than 10 so we have 10 11 and 9 here's an open circle on 10 and let's plug in value for let's plug in 9 is 9 less than 10 that is true so our arrow goes over the nine which is to the left 6 is greater than or equal to over 7 remember it's division so to get rid of division we multiply we're going to multiply both sides by 42 greater than or equal to so 42 in the middle then 43 then 41 before it is closed Circle because it's equal to as well let's plug in value for let's choose the first number 41 remember that 41 goes where is and is on the right is 42 greater than or equal to 41 it is so we're going to draw our line towards the 41 the 41 is the number we plugged in letter 6 is less than 18 divide both sides by six is less than 3 so on our number line 34 and to the left is two pick number to plug in actually it's an open circle on three let's start there I'll pick two to plug it in is 2 less than three that is true so we draw our Arrow over the two and this one was true as well letter divided by 3 so we multiply both sides by three so we get 180 less than or equal to 181 and 179 are our numbers is closed circle on 180 we'll plug in we'll do 179 first 180 less than or equal to 179 is 180 less than or equal to 179 it is not so now let's plug in 181 to make sure 180 less than or equal to 181 that is true statement which means 181 is part of our solution so the arrow goes to the right now there is an additional rule you must know when you are solving inequalities by multiplication and division when you multiply or divide an inequality by negative number the inequality symbol must be reversed in order for the inequality to remain true so we still solve it the same way except when we multiply or divide by negative number we have to actually flip that inequality sign so over here we're going to divide both sides by -2 we get on this side and we still get5 on that side however because divided by that negative have to change this less than symbol to greater than symbol so -5 -4 and6 remember they're negative numbers it is an open circle and let's plug in value for let's try -6 is -6 oops no equal to there is -6 greater than5 well -6 is over here and5 is to the right of it so would have to say this is false so let's try -4 is -4 greater than -5 it is in fact so our arrow goes to the right letter Bops we are going to multiply both sides by3 we'd get on the left and we get -12 on the right we have to flip that symbol so the less than or equal to than symbol becomes greater than or equal to so -2 -11 and -13 it is closed circle on -12 we'll plug in value -13 greater than or equal to -12 that is false so let's try -11 is11 greater than or equal to -12 that is true so our arrow goes to the right over the1 letter want to multiply both both sides by -2 we get on this side and8 on the right side it was less than symbol but because we multiplied by negative number it becomes greater than symbol so is greater than -8 so and 19 it is an open circle on 18 and let's plug in some values is -19 greater than 8 it is not let's plug in -17 is -17 greater than -8 it is so our arrow goes to the right letter divide both sides by -3 and we have to flip the sign greater than or equal to negative / negative is positive and we need to know 223 here so -22 / -3 looks like 7.3 repeating positive so 7 3 repeating so we'll do 7.3 repeating here I'll just count by ones just to make life little simple 8.3 repeating and we'll do 6.3 repeating it doesn't matter what you put above and below as long as it counts by the same increment so here we're counting by whole amounts but we still keep that3 repeating but it doesn't matter if you wanted to put 7.4 repeating here is 7.2 repeating there that would be perfectly fine okay so here we go let's plug in some values let's try 6.3 is 6.3 repeating greater than 7.3 repeating it is not that is false so let's try 8.3 is 8.3 repeating greater than or equal to 7.3 repeating it is okay actually we never drew our Circle here so it's going to be closed circle on 7.3 and the 8.3 was part of our solution so we will draw an arrow to the right letter we're going to multiply by -4 on both sides we get we have to flip the sign and then we get positive 44 so 44 45 and 43 is an open circle let's plug in 43 first is 43 greater than 44 it is not that is false so let's try 45 is 45 greater than 44 it is so our arrow goes to the right letter divide both sides by8 we get POS 7 we flip the sign so POS 7 is greater than or equal to so 7 8 and 6 pick number to plug in remember we're plugging in for is 7 greater than or equal to 6 it is so our circle is closed circle on seven and six is true so our arrow goes to the left