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Easy Way To Solve Absolute Value Inequalities ( Algebra 2)
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Feb 19, 2025
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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