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Testing solutions to absolute value inequalities

CCSS.Math: ,

Video transcript

we have three inequalities here that involve absolute values and then below them we have potential values for X and what I want to do in this video is see which of these potential values for X actually make the inequality true so let's start with this first one in green here on the left it says the absolute value of x is less than the absolute value of negative seven so let's think about which of these X values would make this true and before I even try out the x value so let's see if we can simplify this inequality and the thing that might jump out at you and I encourage you to try these on your own before I work through them what might jump out at you is we know what the absolute value of negative seven is the absolute value of negative seven is how far from zero is negative seven well it's seven to the left it's the distance from negative 7 to 0 is 7 or another way to think about it is the if you the absolute value of any number is always going to be the non-negative version of it so this right over here absolute value of negative 7 is just going to be 7 so an equivalent inequality would be that the absolute value of X needs to be less than 7 so let's see if it's true for x equals negative 8 so if x equals negative 8 then wherever we see the X we put in a negative 8 there so let's see is it an absolute value of negative 8 less than 7 is that true well the absolute value of negative 8 is just going to be 8 so is 8 less than 7 no 8 greater than 7 so that x equals negative 8 does not satisfy the inequality now x equals negative 2 well wherever we see the X let's put a negative 2 so the absolute value of negative 2 needs to be less than 7 what's the absolute value of negative 2 well it's going to be positive 2 is positive 2 less than 7 sure 2 is less than 7 so this this is x equals negative 2 satisfies our inequality the absolute value of negative 2 is going to be less than the absolute value of negative 7 then finally x equals 6 so the absolute value is the absolute value of 6 once again everywhere where I see the X I just put these 6 there x equals six we're going to say the absolute value of six is that less than seven well the absolute value of six is once again just six six is six to the right of zero is 6 less than seven yeah sure 6 is less than 7 so x equals 6 and x equals negative 2 both satisfy the inequality now let's do this one here in this magenta color and once again encourage you to pause the video and try to work through it out on your own so let's try x equals negative 4 so if x equals negative 4 we're going to say the absolute value of negative 4 greater than 5 absolute value of negative 4 well that's just going to be 4 is 4 greater than 5 no 4 is less than 5 so that doesn't work now x equals 3 wherever where we see the X replace that with a 3 the absolute value of 3 is that greater than 5 absolute value of 3 is just 3 so it's 3 greater than 5 no 3 is less than 5 I think you see hopefully you're getting the hang of it so finally if x equals negative 9 the absolute value of negative 9 that knee would need to be greater than 5 absolute value of negative 9 well that's just going to be positive 9 so it's just going to be 9 and is that greater than 5 well sure 9 is greater than 5 so x equals negative 9 satisfies the inequality all right now let's do these ones in this I guess this light purple color the absolute value of x needs to be greater than negative 16 so there's something very interesting about this one we don't even have to look at the choices here can you think of any X for which this would not be true well just think about a little bit the absolute value of a number is it ever going to be negative no the absolute value of a number is going to be 0 or positive it's going to be non-negative so this right over here is 0 or positive or we could call that non-negative 0 or positive so this thing over here is 0 or positive something that is 0 or greater something that is or positive is always going to be greater than a negative number so this is actually true for all X's we don't even have to try them out we can try them out well actually we will try them out just till you see that this is going to be true for all X's because when you take the absolute value if you take the out if X is zero it's going to be zero but for any other value any nonzero value of X the absolute value of it is going to be positive and let's just just see that we could put any X there and this statement is going to be true absolute value of x equals negative fifteen well the absolute value of negative fifteen is that greater than negative sixteen well the absolute value of negative 15 is positive 15 and of course positive 15 is going to be greater than negative 16 a positive is always going to be greater than a negative so this is true if x equals three absolute value of three is that greater than negative 16 a whoops let me write it that way absolute value of three is that greater than negative 16 sure absolute value of 3 is 3 and 3 is positive so it's going to be greater than a negative number so that works and as I said any X would work there and then finally if x equals 9 well absolute value of 9 if x equals 9 is that greater than negative 16 well sure that's just 9 and that is greater than negative 16 even if X was 0 then you would have 0 greater than negative 16 which is absolutely true so any X here works