Absolute Value Inequalities Absolute Value Inequalities
Absolute Value Inequalities
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- I now want to solve some inequalities that also have
- absolute values in them.
- And if there's any topic in algebra that probably confuses
- people the most, it's this.
- But if we kind of keep our head on straight about what
- absolute value really means, I think you will find that it's
- not that bad.
- So let's start with a nice, fairly simple warm-up problem.
- Let's start with the absolute value of x is less than 12.
- So remember what I told you about the
- meaning of absolute value.
- It means how far away you are from 0.
- So one way to say this is, what are all of the x's that
- are less than 12 away from 0?
- Let's draw a number line.
- So if we have 0 here, and we want all the numbers that are
- less than 12 away from 0, well, you could go all the way
- to positive 12, and you could go all the way to negative 12.
- Anything that's in between these two numbers is going to
- have an absolute value of less than 12.
- It's going to be less than 12 away from 0.
- So this, you could say, this could be all of the numbers
- where x is greater than negative 12.
- Those are definitely going to have an absolute value less
- than 12, as long as they're also-- and, x has to
- be less than 12.
- So if an x meets both of these constraints, its absolute
- value is definitely going to be less than 12.
- You know, you take the absolute value of negative 6,
- that's only 6 away from 0.
- The absolute value of negative 11, only 11 away from 0.
- So something that meets both of these constraints will
- satisfy the equation.
- And actually, we've solved it, because this is only a
- one-step equation there.
- But I think it lays a good foundation for the next few
- problems. And I could actually write it like this.
- In interval notation, it would be everything between negative
- 12 and positive 12, and not including those numbers.
- Or we could write it like this, x is less than 12, and
- is greater than negative 12.
- That's the solution set right there.
- Now let's do one that's a little bit more complicated,
- that allows us to think a little bit harder.
- So let's say we have the absolute value of 7x is
- greater than or equal to 21.
- So let's not even think about what's inside of the absolute
- value sign right now.
- In order for the absolute value of anything to be
- greater than or equal to 21, what does it mean?
- It means that whatever's inside of this absolute value
- sign, whatever that is inside of our absolute value sign, it
- must be 21 or more away from 0.
- Let's draw our number line.
- And you really should visualize a number line when
- you do this, and you'll never get confused then.
- You shouldn't be memorizing any rules.
- So let's draw 0 here.
- Let's do positive 21, and let's do a negative 21 here.
- So we want all of the numbers, so whatever this thing is,
- that are greater than or equal to 21.
- They're more than 21 away from 0.
- Their absolute value is more than 21.
- Well, all of these negative numbers that are less than
- negative 21, when you take their absolute value, when you
- get rid of the negative sign, or when you find their
- distance from 0, they're all going to be greater than 21.
- If you take the absolute value of negative 30, it's going to
- be greater than 21.
- Likewise, up here, anything greater than positive 21 will
- also have an absolute value greater than 21.
- So what we could say is 7x needs to be equal to one of
- these numbers, or 7x needs to be equal to one of these
- numbers out here.
- So we could write 7x needs to be one of these numbers.
- Well, what are these numbers?
- These are all of the numbers that are less than or equal to
- negative 21, or 7x-- let me do a different color here-- or 7x
- has to be one of these numbers.
- And that means that 7x has to be greater than or equal to
- positive 21.
- I really want you to kind of internalize
- what's going on here.
- If our absolute value is greater than or equal to 21,
- that means that what's inside the absolute value has to be
- either just straight up greater than the positive 21,
- or less than negative 21.
- Because if it's less than negative 21, when you take its
- absolute value, it's going to be more than 21 away from 0.
- Hopefully that make sense.
- We'll do several of these practice problems, so it
- really gets ingrained in your brain.
- But once you have this set up, and this just becomes a
- compound inequality, divide both sides of this equation by
- 7, you get x is less than or equal to negative 3.
- Or you divide both sides of this by 7, you get x is
- greater than or equal to 3.
- So I want to be very clear.
- This, what I drew here, was not the solution set.
- This is what 7x had to be equal to.
- I just wanted you to visualize what it means to have the
- absolute value be greater than 21, to be more than
- 21 away from 0.
- This is the solution set. x has to be greater than or
- equal to 3, or less than or equal to negative 3.
- So the actual solution set to this equation-- let me draw a
- number line-- let's say that's 0, that's 3, that is negative
- 3. x has to be either greater than or equal to 3.
- That's the equal sign.
- Or less than or equal to negative 3.
- And we're done.
- Let's do a couple more of these.
- Because they are, I think, confusing, but if you really
- start to get the gist of what absolute value is saying, they
- become, I think, intuitive.
- So let's say that we have the absolute value-- let
- me get a good one.
- Let's say the absolute value of 5x plus 3 is less than 7.
- So that's telling us that whatever's inside of our
- absolute value sign has to be less than 7 away from 0.
- So the ways that we can be less than 7 away from 0-- let
- me draw a number line-- so the ways that you can be less than
- 7 away from 0, you could be less than 7, and greater than
- negative 7.
- You have to be in this range.
- So in order to satisfy this thing in this absolute value
- sign, it has to be-- so the thing in the absolute value
- sign, which is 5x plus 3-- it has to be greater than
- negative 7 and it has to be less than 7, in order for its
- absolute value to be less than 7.
- If this thing, this 5x plus 3, evaluates anywhere over here,
- its absolute value, its distance from 0, will
- be less than 7.
- And then we can just solve these.
- You subtract 3 from both sides.
- 5x is greater than negative 10.
- Divide both sides by 5. x is greater than negative 2.
- Now over here, subtract 3 from both sides.
- 5x is less than 4.
- Divide both sides by 5, you get x is less than 4/5.
- And then we can draw the solution set.
- We have to be greater than negative 2, not greater than
- or equal to, and less than 4/5.
- So this might look like a coordinate, but this is also
- interval notation, if we're saying all of the x's between
- negative 2 and 4/5.
- Or you could write it all of the x's that are greater than
- negative 2 and less than 4/5.
- These are the x's that satisfy this equation.
- And I really want you to internalize this
- visualization here.
- Now, you might already be seeing a bit of a rule here.
- And I don't want you to just memorize it, but I'll give it
- to you just in case you want it.
- If you have something like f of x, the absolute value of f
- of x is less than, let's say, some number a.
- So this was the situation.
- We have some f of x less than a.
- That means that the absolute value of f of x, or f of x has
- to be less than a away from 0.
- So that means that f of x has to be less than positive a or
- greater than negative a.
- That translates to that, which translates to f of x greater
- than negative a and f of x less than a.
- But it comes from the same logic.
- This has to evaluate to something that is less than a
- away from 0.
- Now, if we go to the other side, if you have something of
- the form f of x is greater than a.
- That means that this thing has to evaluate to something that
- is further than a away from 0.
- So that means that f of x is either just straight up
- greater than positive a, or f of x is less than negative a.
- If it's less than negative a, maybe it's negative a minus
- another 1, or negative 5 plus negative a.
- Then, when you take its absolute value, it'll
- become a plus 5.
- So its absolute value is going to be greater than a.
- So I just want to-- you could memorize this if you want, but
- I really want you to think about this is just saying, OK,
- this has to evaluate, be less than a away from 0, this has
- to be more than a away from 0.
- Let's do one more, because I know this can be
- a little bit confusing.
- And I encourage you to watch this video over and over and
- over again, if it helps.
- Let's say we have the absolute value of 2x-- let me do
- another one over here.
- Let's do a harder one.
- Let's say the absolute value of 2x over 7 plus 9 is
- greater than 5/7.
- So this thing has to evaluate to something that's more than
- 5/7 away from 0.
- So this thing, 2x over 7 plus 9, it could just be straight
- up greater than 5/7.
- Or it could be less than negative 5/7, because if it's
- less than negative 5/7, its absolute value is going to be
- greater than 5/7.
- Or 2x over 7 plus 9 will be less than negative 5/7.
- We're doing this case right here.
- And then we just solve both of these equations.
- See if we subtract-- let's just multiply everything by 7,
- just to get these denominators out of the way.
- So if you multiply both sides by 7, you get 2x plus 9 times
- 7 is 63, is greater than 5.
- Let's do it over here, too.
- You'll get 2x plus 63 is less than negative 5.
- Let's subtract 63 from both sides of this equation, and
- you get 2x-- let's see.
- 5 minus 63 is 58, 2x is greater than 58.
- If you subtract 63 from both sides of this equation, you
- get 2x is less than negative 68.
- Oh, I just realized I made a mistake here.
- You subtract 63 from both sides of this, 5 minus 63 is
- negative 58.
- I don't want to make a careless mistake there.
- And then divide both sides by 2.
- You get, in this case, x is greater than-- you don't have
- to swap the inequality, because we're dividing by a
- positive number-- negative 58 over 2 is negative 29, or,
- here, if you divide both sides by 2, or, x is less than
- negative 34.
- 68 divided by 2 is 34.
- And so, on the number line, the solution set to that
- equation will look like this.
- That's my number line.
- I have negative 29.
- I have negative 34.
- So the solution is, I can either be greater than 29, not
- greater than or equal to, so greater than 29, that is that
- right there, or I could be less than negative 34.
- So any of those are going to satisfy this absolute value
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