Absolute Value Equations 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 absolute value equations with variables on both sides let's jump into our example where we have the absolute value of 3x - 1 = x + 13 now when we have an absolute value equation with variables on both sides we still solve it the same way basically as as any other absolute value equation we isolate the absolute value and then we write two equations now for this example our absolute value is isolated so now we need to write two equations one equal to a positive and 1 equal to a negative for this equation the absolute value doesn't just equal a number it equals x + 13 don't let that hold you up we can still write two equations 1 equal to a positive and 1 equal to a negative so we have 3x - 1 = x + 13 that's our positive equation or 3x - 1 = NE and then we have x + 13 now you'll notice I have x + 13 Within parentheses and that negative goes on the outside of the parentheses that's because it's applied to everything on that side of the equation so make sure to use parentheses and a negative outside of the parentheses again that way the negative is applied to everything on that side of the equation now that we have our two equations we can solve so let's start with 3x - 1 = x + 13 now remember when we have an equation with variables on both sides we want to get the variable on one side of the equation and then we can look to solve let's start by getting the variable on one side of the equation so let's subtract X from the right side that means we need to subtract X from the left side as well so looking at the right side these X's cancel each other out so we are left with 13 and then on the left side 3x - x we can subtract those like terms that gives us 2X and then we have Min -1 now we have 2X -1 = 13 so we need to look to isolate that variable of X let's undo the subtraction now so let's add one to the left side that means we need to add one to the right side side as well as far as the left side these ones cancel each other out so we have 2X equals and then on the right side 13 + 1 gives us 14 so now we have 2X = 14 so 2 * x = 14 let's undo that multiplication by dividing so divide the left side by two and divide the right side by two these twos cancel each other out so X is now isolated x equals and then on the right side 14 / 2 gives us 7 so x = 7 now let's solve our negative equation and the first thing that we need to do is distribute the negative to what's inside the parentheses so to X and 13 now as far as Distributing this negative let's think of this as-1 so we are Distributing -1 tox and 13 so -1 * X and -1 * 13 that gives us 3x - 1 = --x - 13 our next step is to get the variable on one side of the equation so let's add x to the right side of the equation that means we need to add x to the left side as as well these X's cancel each other out so we have -113 on the right side and then on the left side we can combine those like terms so we have 3x + x that gives us 4X so we have 4x - 1 = -13 so now we need to look to isolate that variable of X let's add one to the left side of the equation that means we need to add one to the right side as well these ones cancel each other out so we have 4X equals and then on the right side -3 + 1 gives us 12 and now we have 4X = --12 so let's divide both sides by four in order to isolate X these fours cancel each other out so X is now isolated x equals and then on the right side -12 / 4 gives us -3 so x = 3 and those are our two solutions 7 and -3 now let's check those solutions by plugging them back into the original equation let's start by plugging in seven so we have the absolute value of 3 * 7 - 1 = 7 + + 13 and then as far as plugging in -3 we have the absolute value of 3 * -3 - 1 = -3 + 13 let's start by checking our solution of 7 so we have the absolute value of 3 * 7 - 1 3 * 7 is 21 - 1 is 20 the absolute value of 20 is 20 so we have 20 equals and then on the right side 7 + 13 is 20 so we have 20 = 20 that is true so we are correct there x equal 7 as far as -3 we have the absolute value of 3 * -3 - 1 3 * -3 is -9 -1 gives us -10 the absolute value of -10 is 10 as far as the right side we have -3 + 13 that equals 10 as well so we have 10 = 10 which is true so we are correct there as well x = -3 so our two solutions 7 and-3 so there you have it there's how to solve and absolute value equation with variables on both sides I hope that helped thanks so much for watching until next time peace