How to Solve Inequalities with Variables on Both Sides | Math with Mr. J

[Music] welcome to math with Mr J in this video I'm going to cover how to solve inequalities with variables on both sides now when we solve inequalities with variables on both sides we want to get the variable on one side of the inequality then we can look to isolate the variable so this is very similar to solving equations with variables on both sides the only thing that's different that we need to be aware of with inequalities remember the inequality symbol flips when multiplying or dividing both sides by a negative let's jump into our example and see how this works we have 2x - 7 is less than 6X + 13 the first thing that we need to do here is get the variable on one side of the inequality so either the left side or the right side now really either side will work we will end up with the same answer either way and we're actually going to do it both ways for this example to show how they both work the first way we're going to look to avoid a negative coefficient so we're avoiding a negative number in front of the X we do that by undoing the 2x so we're looking to undo the smaller coefficient in Val value that's a positive 2x so the inverse operation that we need to use is subtraction so let's subtract 2x from the left side of the inequality whatever we do to one side of an inequality we must do to the other in order to keep it balanced so subtract 2x from the right side as well the two x's on the left cancel each other out so now we have -7 is less than and then on the right side we have 6x - 2x we can subtract those like terms 6x - 2x gives us 4X and then we can bring down + 13 so now we have -7 is less than 4X +3 now our variable X is on one side of the inequality and we can look to solve so we need to isolate the variable now we need to undo the addition the inverse operation of addition that's subtraction so subtract 13 from the right side and then the left side as well these cancel each other out so we are left with 4X and then on the left side -7 - 13 gives us -20 -20 is less than 4X now X is being multiplied by 4 so we need the inverse operation of multiplication that's division so divide the right side by four and the left side by four these fours cancel each other out X is now isolated so we have X is greater than and then on the left we have -20 / 4 that gives us -5 and this is our answer X is greater than -5 so X can be any value greater than -5 and remember we can always rewrite an inequality with the variable coming first all we need to do is write the variable first so X and then make sure the inequality symbol is going the correct way so it needs to open up towards X here x needs to remain greater than -5 either way will work now let's try this the other way and get the variable to the left side which Some people prefer so this will work as well so we have 2x - 7 is less than 6X + plus 13 so let's start by undoing or getting rid of 6X so subtract 6X from the right side and the left side as far as the right side these cancel each other out so we are left with 13 and then on the left side we have 2x - 6X that gives us 4X and then we have - 7 now we have the variable on one side of the inequality let's undo the subtraction next using the inverse operation addition so add seven to both sides these cancel each other out so we are left with -4x is less than and then on the right side 13 + 7 gives us 20 now we have -4x is less than 20 so we need to undo that multiplication by using the inverse operation division so divide the left side by -4 that means we need to divide the right side by4 now remember and this is very important we are dividing both sides by a negative here and as soon as we divide both sides by a negative that inequality symbol flips so these4 fours cancel each other out X is now isolated the inequality symbol has flipped and then on the right side 20 / -4 gives us5 so we get X is greater than -5 that way as well so we can see either way works we get the same answer lastly let's check something greater than -5 and see if it works see if we are correct here let's use 0 0 is greater than -5 so plug in 0 for X we have 2 * 0 - 7 is less than 6 * 0 plus 13 so as far as the left 2 * 0 is 0 and 0 - 7 is is7 now as far as the right 6 * 0 gives us 0 and 0 + 13 gives us 13 -7 is less than 13 so is that true yes so zero did work here we are correct again our final answer X is greater than -5 so any value greater than -5 will work work for X so there you have it there's how to solve inequalities with variables on both sides I hope that helped thanks so much for watching until next time peace