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2. Mutlak Değerin Özellikleri Konusu
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2. Mutlak Değerin Özellikleri Konusu
#matematik #kpss #yks #dgs #dersvideoları #konuanlatımı #tyt #ayt #doğalsayılar #eşitsizlikler #mutlakdeğer
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ÖğrenmeDöküm
00:00Okay everyone, now we've moved on to the properties of absolute value.
00:03Absolute value has seven properties.
00:05We will continue with our first two questions, God willing.
00:08We're on our first question, our first characteristic, folks.
00:11What is our first characteristic?
00:12Of course, that's our initial defining characteristic.
00:15The absolute value of A is equal to the absolute value of minus A.
00:19Right, guys?
00:20He also comes out positive.
00:22If we give this as an example...
00:25Absolute value 3
00:27The absolute value is minus 3.
00:29Right, guys?
00:30He comes out with a negative result.
00:31And as 3, it remains plus 3.
00:34And similarly, if we were to use this as an analogy in an expression...
00:37A minus B
00:37Right, guys?
00:39It's also equal to that.
00:40What's up, guys?
00:41The absolute value is B minus A.
00:44So, this is the negative version of it, folks.
00:46And it comes out positive again.
00:49Now, let's look at our second feature.
00:51In absolute value
00:52A times B
00:54In absolute value
00:56What does that equal, folks?
00:57Absolute value of A
00:59Multiplying the absolute value by B.
01:01We can write this.
01:03To reinforce this with an example...
01:05Here's a simple example.
01:06Minus 2 times 3
01:09In absolute value
01:10It's also equal to that.
01:12What's up, guys?
01:13Absolute value minus 2
01:15What does "multiplied by absolute value" mean, guys?
01:18We write it as 3, plus 3.
01:20It's also equal to that.
01:21What's up, guys?
01:22What is minus 2, guys?
01:23Outwardly positively
01:24So it comes out as 2.
01:253 was already 3.
01:262 times 3
01:27It is 6.
01:30Okay guys, I know that too.
01:32Now let's look at the section expression.
01:35Absolute value, friends.
01:36A divided by B
01:37It's also equal to that.
01:39We can be written separately.
01:41Absolute value of A
01:42B
01:42Absolute value B
01:44Of course, but what are the conditions, friends?
01:46Provided that B is different from zero.
01:49We have to write this.
01:50From where?
01:50If B is zero, this becomes undefined.
01:53Okay, let's give an example.
01:55For example, the absolute value is 7 divided by 3.
01:58What does that equal, folks?
02:01Absolute value divided by 7
02:03We can write the absolute value as 3.
02:06And it comes out as 7/3 anyway.
02:11So, after we've done that, friends...
02:12Let's move on to the expression "taking the upper hand".
02:16Absolute value A to the power of N
02:17It's also equal to that.
02:19The absolute value of A
02:22Above absolute value
02:24We can write it as N.
02:26Let's see the difference.
02:28But they all amount to the same thing, folks.
02:30What is that?
02:31For example, in absolute value.
02:34What does "minus 2" mean, folks?
02:36Let it be the 5th power.
02:38Is it equal, I wonder?
02:40Absolute value
02:41Minus 2 to 5
02:44Yes, it equals.
02:46What do you see when you look at both of them, folks?
02:48What is the 5th power of minus 2, guys?
02:50Minus 32
02:51So, what is the 5th power of minus 2, guys?
02:54Minus 32
02:54But what contains minus 32?
02:57Absolute value minus 32
02:58It's also equal to that.
02:59Again, we can derive it as 2 to the power of 5.
03:02Or we say 32 here, which is equal to.
03:03It's also equal to that.
03:032 to the power of 5 is equal to 5.
03:05Now let's look at the forces in the line, folks.
03:08A to the power of 2N
03:09Absolute value
03:11It's also equal to that.
03:13When you step outside of absolute value
03:14In the form of A to the power of 2N
03:17We can write.
03:18What is our hand here, guys?
03:19An integer.
03:22If we'd like to give an example, friends...
03:26In absolute value
03:27Minus 5
03:28What's up, guys?
03:29Karesi.
03:30It's also equal to that.
03:31What is it equal to, friends?
03:32When you go outside
03:33When it goes beyond absolute value
03:35It happens like this.
03:36Minus 5
03:37What's up, guys?
03:38Karesi.
03:39That's also 5 squared.
03:41It's also equal to that.
03:4225
03:42We'll take it out.
03:45Small, simple, but effective formulas, my friends.
03:48It is used everywhere.
03:49Let's look at our sixth characteristic.
03:52This is known as the triangle inequality, folks.
03:55What is that?
03:56I'm giving you the complete formula.
03:57In absolute value
03:58Absolute value of A
04:00Minus
04:01Absolute value B
04:04I turned off the absolute value.
04:06The difference between the two
04:07Less than or equals, friends.
04:10In absolute value
04:11A plus B
04:12That's also less than or equal to, folks.
04:16Here, also
04:16The inverse triangle inequality also
04:18Inside
04:18It includes.
04:20That's also equal to that, friends.
04:21Absolute value of A
04:22The absolute value of a positive value is B.
04:25We're not going to get into geometry right now, folks.
04:26Just this
04:27Triangle inequality.
04:28Of absolute value
04:33Ok.
04:36Now, when we look at this...
04:37Friends
04:38I'd like to give an example.
04:39For example, A equals 4.
04:41Right, guys?
04:43Our B is equal.
04:44Let it be minus 3.
04:48To see more clearly.
04:51He's a friend too.
04:51In absolute value
04:52Absolute value 4
04:53Minus
04:54Absolute value minus 3
04:55I hung up.
04:57I put them in their places here.
04:59Isn't it?
05:00Less than or equals, friends.
05:02In absolute value
05:034 plus
05:04Minus 3
05:05I'm writing it exactly as it is in the formula.
05:09Less than or equal to.
05:11What's up, guys?
05:13Absolute value 4
05:14Plus
05:16Absolute value minus 3
05:18But is that really the case?
05:20Let's take a look at this.
05:24Absolute value 4
05:25Negative absolute value minus 3
05:27What do these guys do here?
05:30Look now
05:30Absolute value minus 3
05:32It will already come out as a negative result.
05:33It will be 3.
05:34And this is number 4.
05:354 minus 3
05:36Isn't it 1?
05:37Yes, this will be interior number 1.
05:39E has an absolute value of 1.
05:40It already equals 1.
05:41Let's write it like this then.
05:42Let's write the absolute value as 1.
05:44Less than or equal to.
05:45Let's see if it provides that.
05:46We are watching, aren't we?
05:47With an example.
05:484 minus 3
05:49Again, the absolute value is 1.
05:51Less than or equal to.
05:524 plus 3
05:53Why?
05:533 plus 3
05:54Minus 3
05:55How do we derive absolute values from this, guys?
05:57On the negative side.
05:58We'll become familiar with these by now.
06:00Therefore, 4
06:00We say 3 plus 7.
06:04As you can see, how did it turn out, guys?
06:061 less than or equal to
06:071 less than or equal to
06:08The result was 7.
06:09And this inequality is real, folks.
06:12Then this statement of ours is also true.
06:14We will try to make this clearer.
06:16Now, let's move on to our example.
06:18But we have a seventh question, folks.
06:19There's something similar in that, in our seventh characteristic as well.
06:22A is greater than or equal to 0
06:23I'm saying it's about to happen.
06:24Now I remember.
06:25Absolute value x
06:30is equal to a
06:33x equals a
06:35or x equals
06:37It is minus a.
06:38These will also be useful in equations, folks.
06:40How can we do this?
06:42Absolute value x
06:43if it equals 3
06:44x equals 3
06:47or x equals
06:49It's minus 3, folks.
06:54Now let's move on to our question, friends.
06:56In absolute value
06:57absolute value x minus 3 minus
06:58in absolute value
06:593 minus x
07:00minus 7
07:01when the absolute value is closed
07:02equals
07:03We are raising questions.
07:04So, what is it, our friends are asking us.
07:06That's what I wrote.
07:07Now, friends, the absolute value of x is minus 3.
07:12What was it, guys?
07:14You may remember it from here.
07:15a minus b
07:16b minus a equals, right?
07:18Then the absolute value is x minus 3.
07:19the negative expression of this
07:21Isn't 3 minus x equal to...?
07:22Yes.
07:23So what happens then, guys?
07:25Because of this
07:29I'll write it down right away.
07:30Absolute value x minus 3
07:33minus
07:34Instead of 3 minus x, I write x minus 3.
07:36The expression "which is negative"
07:37My friends will come up with the same positive result again.
07:39I don't know the positive result.
07:39Let's not get involved in that now.
07:41Conditions must be set for it.
07:43Greater than, less than x 3, etc.
07:45x minus 3
07:46minus 7
07:48What does that equal, friends?
07:50x minus 3s cancel each other out.
07:51So these cancel each other out.
07:53Inside, the absolute value remains minus 7.
07:56What is the absolute value of -7, friends?
08:00That equals 7.
08:01So our answer is 7.
08:03Now let's look at our next question.
08:04My friends are saying that x is the little girl and her daughter's number is 0.
08:07Absolute value x minus x minus 3
08:10Now, friends, I want to write this down.
08:12x minus the absolute value of x minus 3
08:15minus
08:163 minus x
08:19equals.
08:20Now, friends, if x is less than zero, what is x minus 3?
08:23Let's do this right away.
08:24If x is less than 0
08:25x minus 3
08:27What's up, guys?
08:28What would happen if we subtracted 3 from every angle, guys?
08:30It's less than -3, right?
08:33Minus 3 is ultimately less than 0.
08:36So what will they come out as then, friends?
08:37It will appear as a plus here.
08:40How?
08:41This minus sign will take a minus sign in front of it and
08:43x minus
08:46minus
08:47x minus 3
08:52minus
08:53How will this come out from here, guys?
08:563 minus x
08:57Quite the opposite, friends.
08:59This is because x is less than zero, so this place
09:02Because x is less than 0
09:04if this is 3 minus x
09:05If we multiply by minus, then both sides
09:07-x is greater than 0.
09:09If we add 3, it becomes greater than 3.
09:10So it would be a positive number.
09:11Therefore, the external negative is not the case.
09:12It comes out as a plus and
09:14It won't get any negative marks or anything like that.
09:16So we subtract it as 3 minus x.
09:18In this way.
09:19There's a minus sign in front of it.
09:21Let's write it down.
09:213 minus x.
09:23Now, let's take a look, folks.
09:25Now minus times minus plus.
09:26So what's going on here, guys?
09:28There's no absolute value inside anymore, right?
09:30x plus x minus 3
09:35in absolute value
09:37minus 3 plus x
09:39I've trashed this place, guys.
09:41It's also equal to that.
09:432 x minus 3
09:502x minus 3, friends, plus x minus 3.
09:54Now, when you get to 2 x minus 3, friends...
09:56Because x takes a value less than zero
09:582 times minus 3 will also be less than zero, folks.
10:00It's negative because it's less than zero.
10:02Let's say it's the sum of a negative expression and a negative expression.
10:05It appears as a negative sign, a minus sign, in front of it.
10:06The parenthesis came out as -2x -3.
10:12Plus x minus 3.
10:14It's also equal to that.
10:153 minus 2 x
10:19plus x minus 3.
10:21What does that equal, folks?
10:23I'm writing the result here.
10:25We find the result as minus x.
10:30This much.
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