WEBVTT



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There are two things written on the board, one on

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the left, one on the right.

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What do we call the thing on the left?

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An expression. And what do we call the thing on

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the right? What

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is

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an equation? Anybody know what an equation or what's the

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definition of an equation is?

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Take a wild guess.

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Yeah. Like two expressions brought together by an equal sign.

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Two expressions brought together by an equal sign So

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we definitely have one expression another expression then like an

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equal sign in between okay, we're close Would this be

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an equation definitely not Will this be an

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equation? Definitely not. We definitely need an equal sign.

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And we need an expression on the left and an

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expression on the right.

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Here's the actual definition.

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An equation is a mathematical statement that

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two expressions are equal.

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So an equation is a mathematical statement

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that can be written in many different forms.

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So why is this an equation?

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Whoever said it was.

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Why is it an equation?

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Because it's saying that 2x minus 3 is equal to

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5. Yeah, it's a statement that this expression is equal

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to this expression. Is this an equation?

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Yes. Yes.

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this an equation? see everybody's thumb thumbs up yes thumbs

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down no

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they're

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definitely not equal we agree yeah is

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this a

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mathematical statement

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that

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two Mr. Wittman is 19 years old is

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that a statement?

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That true statement No definitely

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not Okay, so is this a

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statement that? One expression is equal to another expression.

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Yes. Yes, this is an equation Okay

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this is an equation and is the statement true

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yeah yeah this is called a true equation

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this

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is called a false equation Is this equation red?

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Is it true, false, or neither?

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It depends on what the value of x is.

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If x is 3, it would be a true equation.

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If x is any other number, it would be a

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false equation. So this is neither true nor false.

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this is called an open equation.

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Basically, anytime you see a variable inside an equation it's

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an open equation. All equations are

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either true, false,

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or open.

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This is very important.

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Is this an equation?

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Yes, yes X is an expression X is an expression

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is a statement the two expressions are equal

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This contains a variable however.

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This is the rare case that this is not an

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open equation. This would be a what?

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So but most any other time

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like this equation is no longer true always it's

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open okay but almost every time except for special

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cases if an equation contains a

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variable it's an open equation Can

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Can we solve these?

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7th grade? 8th grade?

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Something like that.

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What would be the first step?

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Subtract 5 both sides or add negative 5 both sides.

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Which property allows me to do that?

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Either the addition property or the subtraction property.

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If you add a negative I just call everything the

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addition property of equality.

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And then what's the next step?

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Divide by 2. Okay,

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let

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me come back to

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this. So if the directions were

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solved,

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what does solve mean?

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Make the statement true.

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the statement true. We'll come back to that in a

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second.

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Some people said find the answer.

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What else did we say?

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Find the unknown value.

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Okay, it's a possibility.

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So let's write down the definition.

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to solve

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means

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to find

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the

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solution

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set We'll come back to what

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solutions that means in a second.

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What uh, what is the solution to this equation?

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X equals 3. Uh, it is not x equals 3.

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It is 3. 3

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is the solution.

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So what's the mean to be a solution?

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OK,

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this is very important.

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Here's the definition. a solution

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is

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a number

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that

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when replaced

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for the variable

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it makes

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the equation

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true

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OK, so why is the number 3 a solution?

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Because when we replace it with a variable,

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we no longer have an open equation, we have a

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true equation.

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OK, so a solution is a number that when replaced

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with a variable, it makes the equation true.

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Why is the number 7 not a solution?

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Because when it replaces the variable, it does not make

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the equation true. Okay.

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Okay.

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Solution set,

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the definition is

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a solution set

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is the set of

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all solutions. So

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we still haven't solved yet because we haven't found the

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solution set. Well, we kind of did, but

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our final answer has to be a solution

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set.

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Is x equals 3 the solution?

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Why is x equals 3 not the solution?

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It's still an open equation a solution is

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a number not an equation, okay?

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So the solution is

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is three and the

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solution set

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you

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simply take all the solutions there's only one of them

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here and we put it in these curly brackets

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there's several ways to

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that denotes sets in mathematics.

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If we could list them all, we simply use curly

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brackets and put them all in there separated by comma.

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There's only one here.

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Are these three the same equation?

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They're definitely not the same.

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This is a statement that this expression is equal

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to this expression, which this expression is way different than

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this expression, and this expression is way different.

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And this is way different than that one.

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They're all different equations.

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However they all have something in common.

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What do they have in common?

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The same exact solution set.

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The same exact solution set.

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Okay. Like the addition property of equality says

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you can add the same number to both sides of

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the equation and the new equation is

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equivalent.

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Not the same equations, equivalent.

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So these are all called equivalent equations.

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Equivalent equations

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are

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equations. that have

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the same

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solution

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set.

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Another definition. Okay, so we use the addition property of

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equality to go to here.

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Here we use the division or multiplication property of equality

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which says you can divide both

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numbers by the both sides of the equation by the

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same number and the new equation, it one's not zero

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is equivalent which means that is the same solution set

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okay and once we're here we know the obvious

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solution is three can we plug in three equals three

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okay

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let's try another

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one The directions say solve.

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What does it mean to solve?

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Find the solution set.

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And what is the solution set?

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The set of all solutions.

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The set of all solutions.

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You take them all and you put them in curly

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brackets.

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Okay. And what is the solution?

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A number that when replacing the variable makes the statement

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true. Makes the equation true.

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A number that when replaced for the variable makes the

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equation true. A number that when replaced

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for the variable makes the equation true.

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So solutions, okay first step here is

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Add eight to both sides Next step

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Divide by five or multiply by one fifth same thing

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Is this my solution

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No, we're getting better, I like this.

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What's my solution? Twelve fifths, or two point

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four, something like that.

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And what is the solution set?

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Twelve fifths. So the way we say it, we say

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the set containing

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12

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fifths.

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We write down. It's just curly brackets and put the

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number inside And why is 12 fifth the solution?

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Because when replace for the variable or plug into the

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variable it makes the equation true Why is the number

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zero not a solution?

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Because when you replace a

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variable it does not make the equation true.

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Okay.

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Let's do another one.

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Okay, here we got a very well on both sides.

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So what's our first step?

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extract 3x both

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sides And

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we have four equals seven.

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What does that tell us?

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It's a false equation.

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Good.

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So we use the addition property of equality.

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We end up with a false equation, always false equation.

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So what is the solutions?

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There's none.

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So is there a solution set?

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No.

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I'll leave it. there is.

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Okay, it's simply

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this set.

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The set contains all of its solutions.

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There are no solutions to this equation, but there is

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a solution set. So can we even solve this

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equation? What does it mean to solve?

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Did we find the solution set?

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Yes, we can solve this equation.

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We can't find any solutions but we can solve it.

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Ok, because by the definition we found the solution set.

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In fact this is a very important set.

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Its own name is called the empty set And in

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fact, it's so important we give it its own symbol

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Okay

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It's a circle with a slightly diagonal line through it

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This is also the empty set

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You could either write this as your solution set or

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you could write this as your solution set.

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Simply an empty set.

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In fact, it's one of the most important sets in

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all of mathematics. Is this the empty set?

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Definitely not, this is not empty.

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It contains another set.

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The set containing the empty set is definitely not the

00:14:59.660 --> 00:15:01.480
empty set. So don't ever write this.

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Terrible.

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You can either write this or write this.

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Don't put one inside the other.

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Okay?

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Okay. Let's try a new equation.

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This has lots of variables, what do we get first?

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I like my terms like that

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And now what You could either stop here

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This is always equal to this If you want to

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we could subtract 4x we get this equation

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and it doesn't matter if we do that or not

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and what do we end up with?

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an always true equation so what are the solutions?

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Every single number we know is a solution.

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And what is the biggest set of numbers we know?

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All real numbers. Why is every real number a

00:16:05.580 --> 00:16:08.180
solution? Because when you plug it in for the variable

00:16:08.180 --> 00:16:12.280
it makes the variables in this case makes the equation

00:16:12.280 --> 00:16:16.840
true. Okay, what is the solution

00:16:16.840 --> 00:16:21.000
set? Okay, the solution set the set of real

00:16:21.000 --> 00:16:23.060
numbers. Okay.

00:16:24.580 --> 00:16:27.860
And there's lots of different ways we denote the set

00:16:27.860 --> 00:16:28.780
of real numbers

00:16:30.220 --> 00:16:32.420
One way is we write it out in words like

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that

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Another way is It looks like this have

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we seen that before

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Okay

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That is simply the letter R.

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But in boldface

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That's not IR it's bold face

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are Right when you're in Microsoft Word or whatever how

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do you make something bold?

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We highlight it and press bold

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Okay Before computers you couldn't do that.

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You had to make it bold by hand and so

00:17:10.480 --> 00:17:11.680
back who knows how long

00:17:12.660 --> 00:17:16.980
Okay, they made it bold by making the left side

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Put a line there, okay, that's simply bold face

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are It's just that is the one way we denote

00:17:26.760 --> 00:17:27.720
the set of real numbers.

00:17:27.860 --> 00:17:29.400
We're gonna learn another way.

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Some of you might know this

00:17:31.960 --> 00:17:36.520
we do this all the numbers between negative infinity and

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positive infinity okay we'll typically do this way this is

00:17:41.060 --> 00:17:45.500
called interval notation how many of you guys learned interval

00:17:45.500 --> 00:17:46.880
notation last year

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and not too many we'll learn it uh beginning next

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week i think

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Okay, so some equations have one solution,

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some equations have no solutions,

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and very similar equation all of a sudden we have

00:18:07.450 --> 00:18:09.150
an infinite number of solutions.

00:18:09.950 --> 00:18:14.370
Alright, either way, any equation we can

00:18:14.370 --> 00:18:16.690
solve even if there's no solutions.

00:18:17.490 --> 00:18:18.850
Is this an equation?

00:18:19.650 --> 00:18:20.650
Why is it an equation?

00:18:21.450 --> 00:18:24.110
It's a statement that two expressions are equal.

00:18:24.310 --> 00:18:25.930
As long as you see an equal sign and some

00:18:25.930 --> 00:18:27.490
expression on the left and some expression on the right,

00:18:27.590 --> 00:18:31.070
it's an equation. Anybody know a solution to

00:18:31.070 --> 00:18:34.970
this? Just like say

00:18:34.970 --> 00:18:38.030
like x can go y and y can go like

00:18:38.030 --> 00:18:38.680
5.

00:18:40.810 --> 00:18:41.460
Close.

00:18:42.750 --> 00:18:45.710
So this is fundamentally different because there's two variables.

00:18:47.630 --> 00:18:52.130
So we actually have to alter our definition.

00:18:54.670 --> 00:18:56.430
Where's our definition of equation?

00:18:57.650 --> 00:19:00.250
Here it is. An equation is a mathematical statement that...

00:19:01.490 --> 00:19:02.140
Sorry.

00:19:03.110 --> 00:19:04.110
Definition of a solution.

00:19:04.770 --> 00:19:08.670
A solution is a number that when replaced for the

00:19:08.670 --> 00:19:09.320
variable,

00:19:10.190 --> 00:19:13.770
this is really the only definition for an equation in

00:19:13.770 --> 00:19:17.850
one variable. Okay, when we have an equation in two

00:19:17.850 --> 00:19:22.090
variables, it's a different

00:19:22.090 --> 00:19:23.490
definition. Let's write down

00:19:25.730 --> 00:19:27.130
definition.

00:19:31.150 --> 00:19:33.350
A solution,

00:19:34.710 --> 00:19:37.650
equation,

00:19:40.710 --> 00:19:43.010
and two variables.

00:19:49.210 --> 00:19:51.470
It's not a number.

00:19:52.210 --> 00:19:54.950
We need two different numbers

00:19:54.950 --> 00:19:59.210
okay and

00:19:59.210 --> 00:20:03.730
right for example one comma five

00:20:03.730 --> 00:20:07.470
somebody said or one for the X five for the

00:20:07.470 --> 00:20:10.790
Y is the solution so we had to know which

00:20:10.790 --> 00:20:13.750
number goes to which variable and the way we denote

00:20:13.750 --> 00:20:16.750
that is an ordered pair.

00:20:18.110 --> 00:20:21.610
So a solution to the equation in two variables is

00:20:21.610 --> 00:20:24.410
an ordered pair

00:20:24.410 --> 00:20:28.790
such

00:20:28.790 --> 00:20:29.470
that

00:20:29.470 --> 00:20:33.650
when

00:20:33.650 --> 00:20:34.650
the numbers

00:20:34.650 --> 00:20:39.370
are

00:20:39.370 --> 00:20:43.650
replaced for

00:20:43.650 --> 00:20:46.510
their respective

00:20:46.510 --> 00:20:51.410
variables

00:20:51.410 --> 00:20:55.810
it

00:20:55.810 --> 00:20:57.250
makes the equation true.

00:21:05.870 --> 00:21:08.570
So it's not a number, it's two numbers.

00:21:10.050 --> 00:21:11.450
We should call it ordered pair.

00:21:12.930 --> 00:21:15.790
Such that when the numbers are replaced for the respective

00:21:15.790 --> 00:21:18.110
variables, the x for the x my life or why

00:21:18.110 --> 00:21:19.850
it makes the equation true.

00:21:27.670 --> 00:21:30.450
So 1 comma 5 is a solution.

00:21:30.970 --> 00:21:31.890
Somebody give me another solution

00:21:31.890 --> 00:21:38.150
zero

00:21:38.150 --> 00:21:42.570
comma three well okay

00:21:42.570 --> 00:21:45.090
interesting I like that one half four

00:21:45.090 --> 00:21:48.850
three comma

00:21:48.850 --> 00:21:49.950
nine

00:21:49.950 --> 00:21:54.770
100

00:21:54.770 --> 00:21:55.420
comma

00:21:57.190 --> 00:22:00.270
203

00:22:00.270 --> 00:22:04.810
negative 10 comma

00:22:04.810 --> 00:22:08.010
negative 17

00:22:08.010 --> 00:22:13.170
and

00:22:13.170 --> 00:22:17.470
we could keep going how many solutions are there to

00:22:17.470 --> 00:22:19.470
this equation? Infinite.

00:22:23.570 --> 00:22:28.010
Okay there's no way we're gonna

00:22:28.010 --> 00:22:31.750
write down all the solutions so we're not going to

00:22:31.750 --> 00:22:36.250
do a curly bracket so we can't write

00:22:36.250 --> 00:22:36.900
them all plan.

00:22:40.630 --> 00:22:43.950
So how are we going to write down the

00:22:43.950 --> 00:22:47.470
solution set? Any guesses?

00:22:51.930 --> 00:22:52.580
Okay,

00:22:53.230 --> 00:22:53.880
so I have a question.

00:22:54.590 --> 00:22:57.190
So what would I write?

00:22:57.990 --> 00:23:00.930
So all real numbers?

00:23:01.450 --> 00:23:04.110
And the range is?

00:23:05.430 --> 00:23:06.290
All real numbers.

00:23:08.190 --> 00:23:11.450
So does that mean every ordered pair is a solution?

00:23:12.210 --> 00:23:14.570
No. It's definitely not.

00:23:16.090 --> 00:23:17.830
0, 10 is not a solution.

00:23:19.890 --> 00:23:21.930
So we can't just write down the domain range.

00:23:22.010 --> 00:23:23.990
We'll talk about what domain range means in a minute.

00:23:24.790 --> 00:23:26.370
How am I going to do this?

00:23:27.810 --> 00:23:29.710
Because we know lots of numbers that are not solutions.

00:23:29.710 --> 00:23:32.330
Sorry, lots of ordered pairs.

00:23:37.990 --> 00:23:42.430
This definitely is

00:23:42.430 --> 00:23:43.450
a linear equation.

00:23:48.590 --> 00:23:52.230
Okay, in fact, you've written down the solution set to

00:23:52.230 --> 00:23:54.670
this many, many, many times.

00:23:56.390 --> 00:23:58.470
How do we do it in eighth grade?

00:24:01.490 --> 00:24:02.610
We

00:24:02.610 --> 00:24:07.370
graphed

00:24:07.370 --> 00:24:08.810
the solution set.

00:24:11.410 --> 00:24:12.530
Okay.

00:24:15.050 --> 00:24:17.610
We draw a coordinate plane.

00:24:18.410 --> 00:24:22.990
So when you graph linear equations in two variables in

00:24:22.990 --> 00:24:24.890
eighth grade, and you probably did a bunch last year

00:24:24.890 --> 00:24:28.850
too, you're really just graphing the solution set.

00:24:29.870 --> 00:24:32.930
All the ordered pairs that were replaced for the variable

00:24:32.930 --> 00:24:34.310
make the equation true.

00:24:35.230 --> 00:24:39.570
There's an This line is simply an infinite

00:24:39.570 --> 00:24:43.850
number of ordered pairs Every single ordered pair on that

00:24:43.850 --> 00:24:47.790
line is a solution all of them together is the

00:24:47.790 --> 00:24:48.690
solution set

00:24:49.690 --> 00:24:54.150
Okay We could have graphed the solutions

00:24:54.150 --> 00:24:58.230
for these these are actually also called linear equations but

00:24:58.230 --> 00:24:59.150
only in one variable.

00:24:59.910 --> 00:25:00.650
We could graph it.

00:25:00.910 --> 00:25:03.070
There's only one variable we need only one number line.

00:25:07.310 --> 00:25:10.930
We could put a dot right here on 12

00:25:10.930 --> 00:25:14.690
fifths and we graph the solution set.

00:25:14.970 --> 00:25:16.710
We put a dot on every solution.

00:25:17.490 --> 00:25:19.070
We don't do that because it's pointless.

00:25:19.390 --> 00:25:21.570
I mean it's a lot easier to do this.

00:25:22.630 --> 00:25:26.970
Okay we could have graphed this solution

00:25:26.970 --> 00:25:31.550
set here and we

00:25:31.550 --> 00:25:35.550
put a dot on every single solution which is all

00:25:35.550 --> 00:25:39.810
written numbers but we're gonna do that okay

00:25:39.810 --> 00:25:43.230
but we do graph the solution set to linear equations

00:25:43.230 --> 00:25:47.750
in two variables it's it's really the only way to

00:25:47.750 --> 00:25:50.950
write them all down there are some other notations that

00:25:50.950 --> 00:25:54.710
we don't to worry about that right now

00:25:54.710 --> 00:25:58.670
okay this tells you what is a solution what is

00:25:58.670 --> 00:25:59.570
not a solution

00:25:59.570 --> 00:26:05.770
okay

00:26:05.770 --> 00:26:09.890
so

00:26:09.890 --> 00:26:13.990
did we solve this equation

00:26:13.990 --> 00:26:18.550
yeah why

00:26:18.550 --> 00:26:22.010
do we solve it found the solution set.

00:26:23.470 --> 00:26:25.810
Okay, this is very important.

00:26:26.450 --> 00:26:28.910
We're always either going to start with an expression this

00:26:28.910 --> 00:26:29.930
class or an equation.

00:26:32.230 --> 00:26:32.880
Okay,

00:26:34.450 --> 00:26:35.790
this very first equation,

00:26:37.550 --> 00:26:39.430
sorry, very first Expression I wrote here.

00:26:39.910 --> 00:26:41.450
Can we solve this?

00:26:42.250 --> 00:26:44.730
Why can we not solve It's

00:26:45.850 --> 00:26:49.730
it's not an equation to solve needs to find the

00:26:49.730 --> 00:26:50.720
solution set a

00:26:51.970 --> 00:26:55.670
solution set that all solutions a solutions a number that

00:26:55.670 --> 00:26:58.950
replace for the variable makes the equation true There's no

00:26:58.950 --> 00:26:59.600
equation

00:27:01.050 --> 00:27:03.110
Expressions are neither true or false

00:27:04.090 --> 00:27:08.250
Okay, just is what it is Okay, you cannot solve

00:27:08.250 --> 00:27:12.410
this you can't solve this an equation okay,

00:27:12.410 --> 00:27:15.850
so when we When you look at something around the

00:27:15.850 --> 00:27:17.690
board it's either going to be an expression or an

00:27:17.690 --> 00:27:18.340
equation

00:27:19.290 --> 00:27:23.470
We do totally different stuff Almost always we solve these

00:27:23.470 --> 00:27:27.270
Expressions we can do we can manipulate them You

00:27:27.270 --> 00:27:31.230
could rewrite them.

00:27:31.250 --> 00:27:35.470
You could factor out a number.

00:27:37.170 --> 00:27:41.350
You can change it, manipulate it, simplify it, complexify it,

00:27:41.350 --> 00:27:42.770
but you cannot solve it.

00:27:42.950 --> 00:27:47.430
Very important. A new definition.

00:27:48.230 --> 00:27:52.010
The definition of a relation is simply a set of

00:27:52.010 --> 00:27:52.670
ordered pairs.

00:27:53.830 --> 00:27:55.270
It's called a relation.

00:27:56.070 --> 00:27:57.950
Is this a relation?

00:27:59.890 --> 00:28:02.890
Yes. Yes. Yes, why?

00:28:03.670 --> 00:28:04.670
Because it's a set of ordered pairs.

00:28:05.140 --> 00:28:07.850
Curly brackets mean it's a set.

00:28:07.850 --> 00:28:10.770
and the inside separated by commas are?

00:28:11.070 --> 00:28:12.630
Ordered pairs. Ordered pairs, this set of order

00:28:12.630 --> 00:28:13.280
of pairs.

00:28:14.790 --> 00:28:17.670
Okay. This has a very clear pattern.

00:28:17.990 --> 00:28:19.510
Well, it should be clear.

00:28:21.730 --> 00:28:25.050
1, 3, 2, 5, 3, 7, what's the next one?

00:28:25.750 --> 00:28:28.230
4, 9. 4, 9, so on and so forth.

00:28:28.670 --> 00:28:30.850
If there's a clear pattern, you just put dot dot

00:28:30.850 --> 00:28:31.930
dot, the ellipses.

00:28:32.890 --> 00:28:35.890
Okay, so here this is a relation.

00:28:35.910 --> 00:28:37.190
Why is this a relation?

00:28:37.990 --> 00:28:40.170
Why is this a relation?

00:28:40.970 --> 00:28:43.370
Because it's a set of ordered pairs.

00:28:43.570 --> 00:28:46.890
That's right Because there's the brackets and there's everything inside

00:28:46.890 --> 00:28:49.590
is ordered. This has an infinite of ordered pairs in

00:28:49.590 --> 00:28:54.190
it Okay Okay, there are three ways

00:28:54.190 --> 00:28:58.190
to

00:28:58.190 --> 00:29:02.770
denote relations.

00:29:03.570 --> 00:29:04.450
First of

00:29:04.450 --> 00:29:08.990
all, you list

00:29:08.990 --> 00:29:13.590
the ordered pairs. Just like we did

00:29:13.590 --> 00:29:15.790
here and here. Okay.

00:29:15.790 --> 00:29:17.950
It could be in a table too.

00:29:18.190 --> 00:29:21.670
Doesn't necessarily have to look exactly...

00:29:22.470 --> 00:29:23.120
Okay.

00:29:24.830 --> 00:29:26.850
You can list them like that in the correct...

00:29:26.850 --> 00:29:31.430
This is called roster notation you put them

00:29:31.430 --> 00:29:35.810
in a table you can go 1 5 negative 4

00:29:35.810 --> 00:29:37.390
and put arrows

00:29:37.390 --> 00:29:42.530
2

00:29:42.530 --> 00:29:47.030
ok doesn't really matter as long as you listing your

00:29:47.030 --> 00:29:51.590
ordered pairs. The second way what What else can we

00:29:51.590 --> 00:29:52.370
do with ordered pairs?

00:29:53.170 --> 00:29:56.670
Just like we did up here, we graphed them.

00:29:57.170 --> 00:29:59.790
We graphed the relation.

00:30:00.590 --> 00:30:04.190
Thank ya. Okay, it could be like this or like

00:30:04.190 --> 00:30:07.190
we did earlier. We can have an infinite number of

00:30:07.190 --> 00:30:08.290
ordered pairs.

00:30:09.230 --> 00:30:11.390
That's typically when we graph a relation.

00:30:11.730 --> 00:30:13.230
When there's an infinite number of them.

00:30:14.630 --> 00:30:16.450
And then what's the third way?

00:30:17.850 --> 00:30:19.310
Do you want to take a guess what the third

00:30:19.310 --> 00:30:21.140
way is? This

00:30:23.240 --> 00:30:26.540
Was the solution set this guy, right

00:30:28.140 --> 00:30:31.480
Okay, so the third way is we write down an

00:30:31.480 --> 00:30:32.130
equation

00:30:33.860 --> 00:30:37.940
Okay However, an equation is not

00:30:37.940 --> 00:30:40.780
a set of ordered pairs.

00:30:42.560 --> 00:30:46.800
Okay, so if we write down y

00:30:46.800 --> 00:30:50.920
equals 2x plus 3, this is not a set of

00:30:50.920 --> 00:30:52.900
ordered pairs. But,

00:30:53.940 --> 00:30:56.580
what about this is a set of ordered pairs?

00:30:57.380 --> 00:31:00.640
The solution set. So in equation,

00:31:01.360 --> 00:31:04.160
it's the solution set.

00:31:04.960 --> 00:31:09.180
OK. It's the solution set that is the set of

00:31:09.180 --> 00:31:11.040
ordered pairs. It is the relation.

00:31:11.880 --> 00:31:14.000
So we'll call this a relation

00:31:14.900 --> 00:31:17.600
Technically, it's not the solution set that this is the

00:31:17.600 --> 00:31:18.250
equation

00:31:20.480 --> 00:31:23.940
If the solution set is the set of ordered pairs,

00:31:23.940 --> 00:31:27.180
okay, so those are three ways we denote relations

00:31:28.950 --> 00:31:33.130
Is this an equation?

00:31:34.950 --> 00:31:35.600
Wait, will?

00:31:36.990 --> 00:31:38.370
Sometimes it's a yes.

00:31:38.870 --> 00:31:40.950
Definitely not. Is that an equal sign?

00:31:41.530 --> 00:31:43.750
No. No, definitely not an equation.

00:31:43.990 --> 00:31:44.850
What do we call this thing?

00:31:47.250 --> 00:31:47.900
An inequality.

00:31:48.910 --> 00:31:52.270
Okay. Can we solve inequalities?

00:31:53.970 --> 00:31:55.170
What's the need to solve?

00:31:55.590 --> 00:31:56.310
Find the solution set.

00:31:57.250 --> 00:32:00.170
Okay, it needs to find the solution set.

00:32:00.720 --> 00:32:03.530
What's the definition of a solution to an inequality?

00:32:04.330 --> 00:32:05.910
Actually let's go back.

00:32:05.910 --> 00:32:07.190
what is an inequality

00:32:07.190 --> 00:32:16.830
what

00:32:16.830 --> 00:32:20.790
is

00:32:20.790 --> 00:32:22.070
it

00:32:24.130 --> 00:32:26.570
So, what is the definition of an inequality?

00:32:27.370 --> 00:32:30.430
Umm, close. Not, well, not really.

00:32:31.190 --> 00:32:32.130
A question?

00:32:33.970 --> 00:32:35.230
Sort of.

00:32:36.170 --> 00:32:38.990
It's almost the same definition as equation.

00:32:39.790 --> 00:32:42.790
Or is equation? A mathematical statement that you use questions

00:32:42.790 --> 00:32:44.470
or a less than variable?

00:32:46.110 --> 00:32:49.550
Okay, so instead of equations, it's going to be an

00:32:49.550 --> 00:32:52.190
inequality. We'll write this down in a second.

00:32:52.570 --> 00:32:56.490
It's a mathematical statement that one expression

00:32:56.490 --> 00:33:00.010
is either less than, greater than, less than or equal

00:33:00.010 --> 00:33:01.490
to, greater than or equal to another expression.

00:33:01.690 --> 00:33:02.340
Let me write that down.

00:33:02.330 --> 00:33:06.330
community and inequality

00:33:07.450 --> 00:33:11.970
a mathematical statement that one expression

00:33:12.770 --> 00:33:15.270
is less than

00:33:17.350 --> 00:33:21.950
or greater than or less than or

00:33:21.950 --> 00:33:26.450
equal to or greater than or equal to another

00:33:26.450 --> 00:33:37.650
I

00:33:37.650 --> 00:33:41.650
could have wrote the words less than or greater than

00:33:41.650 --> 00:33:53.150
or

00:33:53.150 --> 00:33:57.730
less than or this is it gonna be a number

00:33:57.730 --> 00:33:58.910
or an ordered pair

00:33:58.910 --> 00:34:03.430
a number

00:34:03.430 --> 00:34:07.330
what because only one variable

00:34:07.330 --> 00:34:11.250
okay

00:34:11.250 --> 00:34:15.530
how many

00:34:15.530 --> 00:34:17.510
solutions does this have

00:34:17.510 --> 00:34:23.710
okay

00:34:23.710 --> 00:34:28.210
is zero a solution

00:34:28.210 --> 00:34:32.230
yes is

00:34:32.230 --> 00:34:33.750
Is 3 a solution?

00:34:34.490 --> 00:34:35.140
Yeah.

00:34:36.610 --> 00:34:37.330
Is 4 a solution?

00:34:37.730 --> 00:34:40.230
Yes. Is 4.5 a solution?

00:34:40.730 --> 00:34:42.570
Yes. Is 5 a solution?

00:34:42.850 --> 00:34:45.110
Yes. Yes. Why is 5 a solution?

00:34:45.290 --> 00:34:46.090
Because it has a variable 2.

00:34:46.870 --> 00:34:51.290
When replaced for the variable, it makes the inequality true.

00:34:52.730 --> 00:34:55.230
Because of the or equal to, this is true.

00:34:56.370 --> 00:34:58.570
Okay. So it definitely has an infinite number of solutions.

00:35:00.470 --> 00:35:03.370
We can write them in something called interval notation.

00:35:03.630 --> 00:35:04.870
We'll learn that probably Monday.

00:35:06.170 --> 00:35:09.170
Right now we can just graph the solution set.

00:35:09.970 --> 00:35:12.490
Since there's only one variable, we need only one number

00:35:12.490 --> 00:35:16.490
line. Okay, and you simply put a dot

00:35:16.490 --> 00:35:20.310
on every single solution.

00:35:21.110 --> 00:35:22.910
You did this last year, correct?

00:35:24.650 --> 00:35:28.970
Solved inequalities. You typically graph a

00:35:28.970 --> 00:35:33.570
solution set. Let's put negative nine.

00:35:34.370 --> 00:35:37.190
What was our first step?

00:35:37.990 --> 00:35:42.450
and then divide by 2 do I have to flip

00:35:42.450 --> 00:35:43.430
the inequality sign?

00:35:44.730 --> 00:35:46.450
No. No. Because

00:35:46.450 --> 00:35:55.290
we're

00:35:55.290 --> 00:35:55.940
not

00:35:56.350 --> 00:35:57.410
No. No.

00:35:58.470 --> 00:36:01.650
All three of these are called equivalent inequalities.

00:36:03.530 --> 00:36:04.180
Okay.

00:36:06.230 --> 00:36:09.310
So we're going to graph the solution set.

00:36:10.110 --> 00:36:12.750
We've got to put a dot on every single solution.

00:36:14.330 --> 00:36:16.070
Is the number 3 a solution?

00:36:16.330 --> 00:36:18.250
Sorry, is the number negative 3 a solution?

00:36:18.590 --> 00:36:19.490
No. No.

00:36:20.890 --> 00:36:21.540
Okay.

00:36:22.970 --> 00:36:25.550
We have to just remove one tiny point.

00:36:26.120 --> 00:36:27.420
Okay, it's very hard.

00:36:27.560 --> 00:36:29.560
If you pull that one tiny point, you'll never see

00:36:29.560 --> 00:36:33.240
that it's gone. so we exaggerated it make a big

00:36:33.240 --> 00:36:37.100
open circle even though

00:36:37.100 --> 00:36:42.200
okay

00:36:42.200 --> 00:36:46.680
again there's gonna be a better way to

00:36:46.680 --> 00:36:48.680
do the solution set but for right now we're just

00:36:48.680 --> 00:36:52.760
gonna graph the solution set okay

00:36:52.760 --> 00:36:56.960
all these definitions are very important what's it

00:36:56.960 --> 00:37:00.500
Equation means we could make the solve with a solution

00:37:00.500 --> 00:37:04.720
set with a solution Put our equivalent equation

00:37:04.720 --> 00:37:06.980
or equation true false and open

00:37:08.200 --> 00:37:12.280
We got to memorize these Know them by heart real

00:37:12.280 --> 00:37:12.930
quick

00:37:13.920 --> 00:37:16.960
We will have a quiz probably in the next week

00:37:16.960 --> 00:37:18.200
on the definitions