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I now want to solve some inequalities that also have

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absolute values in them.

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And if there's any topic in algebra that probably confuses

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people the most, it's this.

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But if we kind of keep our head on straight about what

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absolute value really means, I think you will find that it's

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not that bad.

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So let's start with a nice, fairly simple warm-up problem.

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Let's start with the absolute value of x is less than 12.

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So remember what I told you about the

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meaning of absolute value.

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It means how far away you are from 0.

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So one way to say this is, what are all of the x's that

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are less than 12 away from 0?

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Let's draw a number line.

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So if we have 0 here, and we want all the numbers that are

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less than 12 away from 0, well, you could go all the way

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to positive 12, and you could go all the way to negative 12.

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Anything that's in between these two numbers is going to

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have an absolute value of less than 12.

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It's going to be less than 12 away from 0.

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So this, you could say, this could be all of the numbers

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where x is greater than negative 12.

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Those are definitely going to have an absolute value less

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than 12, as long as they're also-- and, x has to

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be less than 12.

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So if an x meets both of these constraints, its absolute

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value is definitely going to be less than 12.

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You know, you take the absolute value of negative 6,

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that's only 6 away from 0.

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The absolute value of negative 11, only 11 away from 0.

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So something that meets both of these constraints will

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satisfy the equation.

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And actually, we've solved it, because this is only a

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one-step equation there.

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But I think it lays a good foundation for the next few

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problems. And I could actually write it like this.

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In interval notation, it would be everything between negative

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12 and positive 12, and not including those numbers.

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Or we could write it like this, x is less than 12, and

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is greater than negative 12.

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That's the solution set right there.

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Now let's do one that's a little bit more complicated,

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that allows us to think a little bit harder.

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So let's say we have the absolute value of 7x is

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greater than or equal to 21.

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So let's not even think about what's inside of the absolute

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value sign right now.

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In order for the absolute value of anything to be

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greater than or equal to 21, what does it mean?

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It means that whatever's inside of this absolute value

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sign, whatever that is inside of our absolute value sign, it

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must be 21 or more away from 0.

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Let's draw our number line.

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And you really should visualize a number line when

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you do this, and you'll never get confused then.

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You shouldn't be memorizing any rules.

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So let's draw 0 here.

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Let's do positive 21, and let's do a negative 21 here.

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So we want all of the numbers, so whatever this thing is,

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that are greater than or equal to 21.

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They're more than 21 away from 0.

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Their absolute value is more than 21.

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Well, all of these negative numbers that are less than

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negative 21, when you take their absolute value, when you

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get rid of the negative sign, or when you find their

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distance from 0, they're all going to be greater than 21.

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If you take the absolute value of negative 30, it's going to

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be greater than 21.

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Likewise, up here, anything greater than positive 21 will

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also have an absolute value greater than 21.

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So what we could say is 7x needs to be equal to one of

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these numbers, or 7x needs to be equal to one of these

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numbers out here.

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So we could write 7x needs to be one of these numbers.

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Well, what are these numbers?

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These are all of the numbers that are less than or equal to

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negative 21, or 7x-- let me do a different color here-- or 7x

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has to be one of these numbers.

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And that means that 7x has to be greater than or equal to

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positive 21.

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I really want you to kind of internalize

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what's going on here.

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If our absolute value is greater than or equal to 21,

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that means that what's inside the absolute value has to be

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either just straight up greater than the positive 21,

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or less than negative 21.

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Because if it's less than negative 21, when you take its

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absolute value, it's going to be more than 21 away from 0.

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Hopefully that make sense.

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We'll do several of these practice problems, so it

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really gets ingrained in your brain.

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But once you have this set up, and this just becomes a

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compound inequality, divide both sides of this equation by

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7, you get x is less than or equal to negative 3.

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Or you divide both sides of this by 7, you get x is

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greater than or equal to 3.

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So I want to be very clear.

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This, what I drew here, was not the solution set.

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This is what 7x had to be equal to.

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I just wanted you to visualize what it means to have the

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absolute value be greater than 21, to be more than

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21 away from 0.

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This is the solution set. x has to be greater than or

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equal to 3, or less than or equal to negative 3.

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So the actual solution set to this equation-- let me draw a

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number line-- let's say that's 0, that's 3, that is negative

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3. x has to be either greater than or equal to 3.

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That's the equal sign.

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Or less than or equal to negative 3.

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And we're done.

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Let's do a couple more of these.

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Because they are, I think, confusing, but if you really

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start to get the gist of what absolute value is saying, they

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become, I think, intuitive.

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So let's say that we have the absolute value-- let

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me get a good one.

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Let's say the absolute value of 5x plus 3 is less than 7.

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So that's telling us that whatever's inside of our

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absolute value sign has to be less than 7 away from 0.

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So the ways that we can be less than 7 away from 0-- let

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me draw a number line-- so the ways that you can be less than

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7 away from 0, you could be less than 7, and greater than

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negative 7.

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Right?

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You have to be in this range.

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So in order to satisfy this thing in this absolute value

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sign, it has to be-- so the thing in the absolute value

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sign, which is 5x plus 3-- it has to be greater than

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negative 7 and it has to be less than 7, in order for its

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absolute value to be less than 7.

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If this thing, this 5x plus 3, evaluates anywhere over here,

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its absolute value, its distance from 0, will

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be less than 7.

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And then we can just solve these.

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You subtract 3 from both sides.

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5x is greater than negative 10.

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Divide both sides by 5. x is greater than negative 2.

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Now over here, subtract 3 from both sides.

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5x is less than 4.

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Divide both sides by 5, you get x is less than 4/5.

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And then we can draw the solution set.

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We have to be greater than negative 2, not greater than

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or equal to, and less than 4/5.

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So this might look like a coordinate, but this is also

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interval notation, if we're saying all of the x's between

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negative 2 and 4/5.

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Or you could write it all of the x's that are greater than

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negative 2 and less than 4/5.

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These are the x's that satisfy this equation.

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And I really want you to internalize this

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visualization here.

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Now, you might already be seeing a bit of a rule here.

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And I don't want you to just memorize it, but I'll give it

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to you just in case you want it.

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If you have something like f of x, the absolute value of f

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of x is less than, let's say, some number a.

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Right?

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So this was the situation.

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We have some f of x less than a.

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That means that the absolute value of f of x, or f of x has

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to be less than a away from 0.

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So that means that f of x has to be less than positive a or

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greater than negative a.

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That translates to that, which translates to f of x greater

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than negative a and f of x less than a.

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But it comes from the same logic.

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This has to evaluate to something that is less than a

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away from 0.

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Now, if we go to the other side, if you have something of

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the form f of x is greater than a.

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That means that this thing has to evaluate to something that

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is further than a away from 0.

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So that means that f of x is either just straight up

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greater than positive a, or f of x is less than negative a.

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Right?

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If it's less than negative a, maybe it's negative a minus

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another 1, or negative 5 plus negative a.

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Then, when you take its absolute value, it'll

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become a plus 5.

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So its absolute value is going to be greater than a.

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So I just want to-- you could memorize this if you want, but

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I really want you to think about this is just saying, OK,

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this has to evaluate, be less than a away from 0, this has

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to be more than a away from 0.

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Let's do one more, because I know this can be

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a little bit confusing.

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And I encourage you to watch this video over and over and

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over again, if it helps.

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Let's say we have the absolute value of 2x-- let me do

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another one over here.

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Let's do a harder one.

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Let's say the absolute value of 2x over 7 plus 9 is

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greater than 5/7.

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So this thing has to evaluate to something that's more than

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5/7 away from 0.

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So this thing, 2x over 7 plus 9, it could just be straight

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up greater than 5/7.

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Or it could be less than negative 5/7, because if it's

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less than negative 5/7, its absolute value is going to be

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greater than 5/7.

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Or 2x over 7 plus 9 will be less than negative 5/7.

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We're doing this case right here.

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And then we just solve both of these equations.

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See if we subtract-- let's just multiply everything by 7,

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just to get these denominators out of the way.

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So if you multiply both sides by 7, you get 2x plus 9 times

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7 is 63, is greater than 5.

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Let's do it over here, too.

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You'll get 2x plus 63 is less than negative 5.

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Let's subtract 63 from both sides of this equation, and

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you get 2x-- let's see.

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5 minus 63 is 58, 2x is greater than 58.

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If you subtract 63 from both sides of this equation, you

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get 2x is less than negative 68.

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Oh, I just realized I made a mistake here.

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You subtract 63 from both sides of this, 5 minus 63 is

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negative 58.

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I don't want to make a careless mistake there.

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And then divide both sides by 2.

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You get, in this case, x is greater than-- you don't have

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to swap the inequality, because we're dividing by a

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positive number-- negative 58 over 2 is negative 29, or,

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here, if you divide both sides by 2, or, x is less than

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negative 34.

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68 divided by 2 is 34.

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And so, on the number line, the solution set to that

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equation will look like this.

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That's my number line.

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I have negative 29.

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I have negative 34.

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So the solution is, I can either be greater than 29, not

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greater than or equal to, so greater than 29, that is that

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right there, or I could be less than negative 34.

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So any of those are going to satisfy this absolute value

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inequality.

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