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Robert McKeown: Here's our first question where

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we have an absolute value. And the question

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says, graph the solution to the inequality on

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the number line, or what I would call the real

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number line. When I have a question like this, I

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like to transform the inequality into an

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equality. So I've got the absolute value of x

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plus three is equal to four. And my first

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question to myself is, what values of x satisfy

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the quality. And looking at this, I can see that

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if x is equal to one, then x plus three is equal

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to four.

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with absolute values, there's often more than

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one solution. And when we talk about functions,

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we're going to see that with some special

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functions later, but there's also the fact that

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we could have more than one solution. So looking

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at this, if x, and I'm just sort of plugging

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numbers into my mind, and guess saying, if x is

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equal to negative seven, then x plus three is

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also equal to four. Because I have, of course,

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its absolute value. So I have negative seven

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plus three, I've got the absolute value of

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negative four is equal to four. Right, so

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they're using the absolute value operator.

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Now, let me clear this up a little bit. Now

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notice that if x is greater than one,

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the equation or the inequality fails to halt.

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And if x is less than negative seven, it also

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fails to hold. I should also point out that the

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way I've written the axis is not entirely

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correct. x should be greater than or equal to

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one, and x should be less than or equal to

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negative seven. Because if x is actually equal

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to one that this inequality doesn't hold,

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because it's less than sign, not a less than or

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equal to sign. Now looking at the real number

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line, I could have this, you know, going off to

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infinity over here, and off to negative oops,

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not at eight. That's not what I want.

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I can have one here, and a negative zero, or

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sorry, excuse me a negative seven there. And

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as long as x falls between these two values, the

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absolute inequality holds. So how am I going to

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answer this question on ALEKS? Here's the

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question on ALEKS. I'm going to use the circle

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that is hollow. And I'm going to identify the

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two points of interest. The two points of

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interest are one and negative seven. That I'm

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going to press this line icon and it looks like

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it's automatically filled in the correct

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interval where this inequality is satisfied. And

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now I will. Oops. And now I'm plus the check

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Let's see if we get the right answer. And we

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did. So I have to remember that it's an open

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interval, or open set. And so we're using the

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dot that's hollow on both the upper value and

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the highest value and the lowest via

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this question is to keep you working on your

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algebra skills when it comes to doing this type

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of question where you have an expression, and

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you're being asked to solve this equation for y

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two, and what does that mean, they want us to

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isolate y two on one side of the equation. A

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economists, undergraduate majoring in economic

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economics, you're going to be asked to do this

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all the time. If you can do it well, if you can

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do it easily. If you practice it so that you can

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do it before the test you're good at have a

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great undergraduate career, you're going to be

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very relaxed, and everything is going to be

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easy. So let's go ahead and let's try and solve

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this equation for y two. The first thing I want

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to do is I want to get rid of this denominator.

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And how am I going to do that, I'm going to

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multiply both sides of the equation by x two

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minus x one. And so I'm going to have m times x

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two minus x one is equal to y two minus y one.

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And I'm pretty much there. I'm going to add y

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one to both sides of the equation.

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And there's my answer. Now, let's go ahead and

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put it into ALEKS and see if it's correct. So I

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am here on ALEKS. You can see the question is

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the exact same question we solved on the slides.

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So let's go ahead and input the answer that we

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came up with got y to z equal to m. x, and I'll

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use this button here. And then I press the right

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arrow to move over x. And press that button

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again, one and then the right arrow on my

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keyboard to get it to go like that. And the last

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bit is to add y one. And let's see if we have

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the right answer. And we do we have the right

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answer. So pretty straightforward. Just showing

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you that really, you can multiply both sides of

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the equation by more than one term, right, we

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multiply both sides of the equation by x two

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minus x one. As long as we do that properly,

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there's no problems with doing that.

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I'd like you to take a look at the question in

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front of you. It says at the top for each

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equation, choose the statement that describes

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its solution, if applicable, give the solution.

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Now you can see that you've got three different

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boxes to choose from. One is that there's no

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solution. The second box as well. Maybe the

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unknown variable has a value. That's the

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solution, something like w is equal to two. And

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the next option is that all real numbers are

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solutions. So let's take a let's go back to the

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slides now. And let's talk about the possible

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answers to a question like this. So looking at

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your slides, I'll let you sort of read it

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yourself. But you can see that when we solve an

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equation like we were given on ALEKS, there are

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really three possibilities that are gonna come

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up. The first possibility is that we're going to

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get a contradiction. We're gonna get something

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like zero was equal to four, or three is equal

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to negative two, something like that. If we end

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up with that result, that means there is no

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solution. There's no actual solution to that

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equation. Two is the most common one. That's the

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one that you're used to and you expect is that

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the answer is going to be something like x is

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equal to two and x is equal to eight or

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something like that. And you've seen lots of

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questions like that. The final possibility is

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that You're going to get an identity, something

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like zero is equal than zero, or three is equal

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to three. What that means is that any value for

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the unknown variable, so suppose the unknown

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variables x, so any value of x will satisfy that

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equation, the value of x does not matter. And

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any value of x will solve the equation. And

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that's what it says here, all real numbers are

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solutions. Now, here's the problem that we're

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given. So the first thing I'm going to do, I'm

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going to start with the equation on the left,

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and I'm going to solve for w, and I'm going to

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see what I get. So let's multiply four inside

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the bracket. So I've got four w, minus four

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minus one is equal to four w minus six. Now

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notice, if I subtract four w on both sides of

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the equation, they're just going to cancel each

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other out. Or, more specifically, they're going

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to sum to zero. Now I've got minus five is equal

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to minus six, Hmm, well, I know that can't be

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true. So unless I've made a mistake, this is a

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contradiction. And there is no solution.

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Now let's take a look at the equation on the

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right hand side, I'm going to start with the

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same process, I've got four x plus four, plus x

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is equal to three x minus six plus two. So I'm

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working out the bracket. Now I'm going to

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collect like terms, and I've got five x plus

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four is equal to three x minus four. And if I

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subtract four on both sides of the equation, and

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I subtract three x on both sides of the

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equation, I get two x is equal to minus eight,

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and x is equal to negative four. And so you can

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see that there is a unique solution, there is

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one solution to the equation on the right

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equation on the left, there is no solution.

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Our last question is short. And to the point,

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we're going to look at an absolute value

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equation, this time similar to the one we just

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did. There could be more than one solution,

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there could be one solution. Or there could be

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no solution at all. So to solve this, I'm going

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to isolate the absolute value of u. And I can do

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that by dividing both sides by 2am. I done? Is

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there a solution? Well, one thing to remember is

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that absolute value denotes distance. On the

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real number, line. Negative distance

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negative distance is impossible.

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This means that the absolute value of u, which

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is equal to negative two has no solution. It's

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not possible for an absolute value to be less

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than zero. And so there's no solution here. I

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wanted to illustrate to you an example, with

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absolute value, where there is no solution so

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the solution is not Yeah, there's no you that

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satisfies an answer of negative two