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Have you ever been confused about how to solve absolute value inequalities? You're not alone! Many students find absolute value inequalities difficult to understand. However, once you know a few key concepts, they can be solved relatively easily. In this video, we'll go over some of the basics of solving absolute value inequalities. By the end, you should have a better understanding of how to tackle these types of problems. Let's get started!
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so let me redo it all over again so
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basically we have three cases right
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so if I have this inequality ax plus b
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absolute value is greater than or equal
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to C this is what you're gonna have
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you're gonna have two cases number one
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ax plus b is going to be less than or
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equal to negative C or
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a X plus b is going to be more than or
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equal to C
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and then we can solve it I'll give you
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an example if you have 2X
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plus 5
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is more than or equal to four
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you split this into two okay now watch
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the pair that you pay attention to the
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the inequality design when it's more
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than or equal you have to split that
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into two
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ax plus B has to be less than or equal
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to negative C or a X Plus B has to be
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more than or equal to C and this is it
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this implies a union meaning you're
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gonna put the two sets together
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so if you were to solve this right John
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you say anything
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2X
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plus 4 is less than five
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is less than or equal to negative four
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or
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two x
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plus five
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is more than or equal to four and then
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you can solve for x right that's going
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to give us here so
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no this is negative this is positive
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okay so here you're gonna have two x is
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less than or equal to negative nine so
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just subtracting five
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and then X is
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less than or equal to negative nine over
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two second case
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two x is more than or equal to negative
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one
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and X is more than equal to negative one
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half right now once you get to this
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point you just have to put your
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equations your inequalities on the
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normal line right put it it's not an
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intersection it's a union you're putting
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them together
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so if I draw my number line here
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negative nine half is here negative one
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half is here so since this is more than
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or equal to negative one-half going to
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Infinity
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close Circle this way this is less than
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12 circle this way if you want to put
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this in the in a
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edible notation is going to be negative
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Infinity to negative nine half close
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bracket Union
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negative one half to positive Infinity
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that's going to be your solution right
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that's if absolute value of a X plus b
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is greater than or equal to C
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that's when that happens does that make
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sense any questions
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stop doing that especially with me on it
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you know what
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next case
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a X plus b absolute value is less than
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or equal to C how do you solve this
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right so in this case since this is less
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than or equal a X plus b is going to be
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between C and negative C pay attention
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so watch the inequality of the time when
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this is less than or equal you just have
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to split them like this okay and we
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solve it so example
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and if I have 2X minus 5 is less than or
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equal to 5.
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I'm going to strip this right I'm gonna
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put this in between it's going to have
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boundaries so 2x minus 5 is going to be
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between 5
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and negative five that's my a that's my
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B that's my C
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and I'm going to solve for x same way as
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five
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add five
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so if you have zero is less than or
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equal to two x that's less than or equal
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to 10 and then divide it by two
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five two by two I have zero it's less
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than or equal to X less than or equal to
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five and that's going to be zero to five
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okay this is what you do now that's only
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if the absolute value of the expression
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is less than or equal to
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your number okay in the other case you
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split that you have or which is a union
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all right now special case what if I
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have this
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I want you all to think about it
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how would I solve this absolute value of
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x minus 5 is greater than or equal to
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negative two
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how did you solve this
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yeah next solution actually there is a
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solution what's the solution yeah plus
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five nope
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um
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would be x minus five
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is greater than negative two when it's
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less than positive two well watch here
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the number is what negative right yeah
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in all other cases the number was what
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positives right
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what do we know about Oxford value is he
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always positive
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so what's the solution
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no
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oh
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right this is always going to be true
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absolute value of x minus 5 is always
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going to be greater than or equal to
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negative two no matter what number you
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put in here
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all real real numbers this is always
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going to be true right is that only when
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the the number after the greater than
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sign is negative when it's negative
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that's what that's what uh happens okay
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if because absolute value is always what
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positive the absolute value of a number
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is always positive so that means it's
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always going to be greater than or equal
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to negative 2. that's always going to be
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the case right right
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any questions
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now what if I change this
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what if I have this 12 meter solution
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here
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think in terms of the same way yeah
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no solution why
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everything that's in the afternoon
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always positive right positive
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yeah so this will be no solution because
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that's never gonna happen right no
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solution because absolute value is
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always positive
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all right
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um
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is less than a conference like positive
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number so you cannot have it can never
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happen it can
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it can last a negative number does that
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make sense because that's not going to
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happen right absolute value of the
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number is always positive so you can
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never be less than negative two what so
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that would be a new solution
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foreign
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things you guys can figure it out
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