 So absolute value expressions give us absolute value equations So how do we solve absolute value equations? There are two approaches first There's the easy way do a little geometry and then do a little algebra But if you don't like geometry if you don't like drawing pictures We can always do things the hard way which is to use algebra only We'll do things the easy way first Doing things the easy way requires remembering two important concepts first the absolute difference a Minus b is the distance between a and b on the number line and Second we're not always going to get an expression a minus b So it's useful to remember that a plus b is the same as a minus negative b For example, let's try to solve this and again how you speak influences how you think so rather than reading this as Absolute value of x minus 5 equals 3 It's better to read this as the absolute difference between x and 5 is 3 So remember absolute difference a minus b is the distance between a and b on the number line And so we can read this absolute difference x minus 5 is the distance between x and 5 The equation says that that distance will be equal to 3 So we're looking for something that is 3 away from 5 so we'll draw our number line and locate 5 and Wherever x is it's 3 away So x will either be 3 more than 5 or 3 less than 5 So let's draw that we'll put x up here at 3 more than 5 But we could also have x down here at 3 less than 5 So what's x if x is up here then x is equal to 8 But if x is down here, then x is equal to 2 And so we have our solutions x equals 2 or x equals 8 How about the absolute difference between 5 and x is equal to 3? So again this absolute difference between 5 and x is the distance between 5 and x on the number line The equation says that that distance is 3 so we're looking for something that's 3 away from 5 But that's what we just did x will be 3 more than 5 or 3 less than 5 So x equals 2 or x equals 8 How about something like this? One of the important things is that if you read this equation you read it as the absolute difference of 3x plus 2 At which point you say hey wait a minute. That's not a difference. That's a sum Well, actually it is it's just not obviously a difference And so this relies on that property of the real numbers a plus b is the same as a minus negative b and so this 3x plus 2 we can rewrite this as 3x minus negative 2 and So now this is an absolute difference and we can read it as saying that the distance between 3x and negative 2 is 8 so let's draw our number line and We know that 3x will be 8 more than negative 2 or 8 less than negative 2 so wherever negative 2 is How about here? 3x could be up here at 8 more than negative 2 Or 3x could be down here at 8 less than negative 2 So if we're up here at 8 more than negative 2 we must be at 6 And if we're down here at 8 less than negative 2 then we must be at negative 10 and So that tells us 3x equals 6 or 3x equals negative 10 Again, if it's not written down it never happened. So let's record this information Since this is an algebraic problem We can't avoid doing at least a little bit of algebra and so we have two equations 3x equals 6 or 3x equals negative 10 so we'll solve these equations separately And we get our solutions x equals 2 or x equals negative 10 thirds Or let's try to solve this equation We have the absolute difference between x and 5 is the distance between x and 5 The equation says we want this distance to be negative 2 But distance can't be negative So that means that this equation has no solution and we'll record that conclusion