00:00 Hello, my name is Oyase Sunday, I will be taking you to JSS2 mathematics today, and

00:19 we are looking at an interesting topic that says an introduction to the linear inequalities.

00:25 Now we are having an introduction to the linear inequalities, what does that mean?

00:29 It means that an inequality, there are two signs in mathematics, we have the equation

00:36 sign and the inequality sign, so it means that your algebraic expression can be in equation,

00:46 if it's not given in equation, the next form, the expression can be given, it's either

00:51 on a less than or on a greater than.

00:55 Also equation algebraic expression can be expressed in an equation or in the inequality

01:02 form.

01:03 So, this class is specially dedicated to the solving algebraic expression using what the

01:09 less than, greater than, less than equal to, greater than equal to.

01:15 Okay, the objective of the class is to identify linear inequality in one variable, to solve

01:22 linear inequality in one variable, that means all what we are looking at here is going

01:27 to be one variable.

01:28 We have linear inequality in two variables, those are the subsequent videos, but this won't

01:34 be concentrating on one variable, representing solution of linear inequality in one variable

01:40 on a number line, and solve what problem involving linear inequality in one variable.

01:46 So, all our attention here will be directed towards one variable, that is when you have

01:52 x or y or p or q.

01:56 Okay, move up, an algebraic inequality, an algebraic inequality is a mathematical sentence

02:03 that includes a not equal to sign, like I said earlier, greater than or a less than symbol,

02:09 to show the algebraic expression and not equal.

02:12 So, we have two signs, greater than and have less than, you know, in primary school, remember

02:19 those days, we use our hands to indicate greater than and less than, okay, so that is the form

02:26 to which the mathematical expression will be taking the form.

02:31 So, if I have greater than and less than, so I will have, if root 2, I will have, I have

02:40 for example, 4 less than 8 plus 2, or 22 minus 7 is greater than, so when I say 4 is less

02:52 than 8 plus 2, okay, it means that 22, I can also say 22 is minus 7 is greater than 5, okay,

03:05 so you could see this form.

03:07 Now, the inequality I usually represented on a number line, number line, so it becomes

03:16 very simple to represent your inequality on a number line.

03:20 But looking through, like I said, it has to be on the one variable, so everything concentration

03:26 here is going to be on one variable, so if I have 2x plus 3, less than 4, all my unknown

03:36 is going to be what x, remember x, remember your number line, I told you, we're going to

03:42 be representing your number line, so that's what we're working on, so let's pick question

03:45 number 1.

03:46 Question number 1 says, if x equal to 3, determine whether the inequality 3x minus 5 is less

03:55 than x plus 4 is true, now one is saying that they've given you the value of x, this becomes

04:00 so easy because they've given you the value of x, what they just expected you to do is substitute

04:05 x, the value of x into the equation, wherever you see the x, put the value of x, so at the

04:11 end of the day, what you get, is it going to be less or is it going to be greater?

04:17 So, what put it in, I have question number 1, so I have question 1, I have x equal to

04:25 3, so put in the value, I have 3x minus 5, less than x plus 4, so I have 3, open bracket,

04:36 what's my x, 3, minus 5, less than, what's my x, 3, plus 4, 3 times 3, 9 minus 5, less than

04:47 7, so 9 minus 5, that gives me what, 4, less than 7, so is this statement true, therefore

04:55 it's less than 7, is it true, is it true, so you have to say if it is true, yes, if it is

05:03 false, no, so 4 is 4 actually less than 7, yes, as that means the statement is true, okay,

05:13 but there's another part of inequality that is called the less than equal to sign, that

05:23 is when you have, you know we said they have 4 signs, I usually take note of 4 signs of inequality.

05:31 Now, we have 4 signs of inequality, I have the less than sign, that's the less, I have

05:47 the less than equal to sign, less than equal to, and I have the greater than sign, greater

05:59 than sign, I have the greater than equal to sign, I have the greater than equal to, so these

06:11 are the 4 signs we are going to be working with, the 4 signs, but like I said earlier you

06:17 represent your on the number line, we stick on the number line where we will be asked to solve

06:22 questions and put them on the number line, so question number 2 says solve the inequality

06:31 and check for your stack, check the solutions, remember for the 4th one I solved I had to check

06:36 whether it was true or not, so we have the question here is that's question number 2, says

06:45 I have minus 6, open bracket x minus 6 plus bracket plus 4, less than equal to minus 8,

06:56 x minus 10, now if you look to what you need to do first, remember open your bracket, open

07:03 the bracket, so that the first step bracket, remember your both marks, bracket, open your

07:09 bracket, come first, so this time this that give us minus 6x, minus times plus, minus times

07:17 minus gives us plus, 6 times 6 that gives 36 plus 4, less than equal to minus 8, x minus 10,

07:30 what do you do, you collect the light time, so I have minus 6x, that give us 40, that's

07:37 36 plus 4, less than equal to minus 8, x minus 10, okay, so we have categorically, what do

07:47 you do next, you find the additive inverse, remember the additive inverse, any value you want

07:53 to eliminate, I'm eliminating plus 40, so the additive inverse plus 40 is minus 40, I'm

07:58 letting minus 8x, the additive inverse is plus 8x, so my additive inverse give me 6x plus 40,

08:06 minus 40, apply it to both sides, then minus 10 plus 40, that I have minus 6x, less than

08:24 equal to minus 8x, then minus 40 plus minus 10 plus 40, minus 10 plus 40, minus 10, okay,

08:41 minus 40, because I am eliminating, so I have minus 10 minus 40, so that gives me minus 50,

08:51 now additive inverse that's minus 6, okay, plus 8, you're taking it out, that's less than

08:57 equal to minus 8x, minus 50, minus plus 8x, remember, so this can take out this effectively

09:10 out of the box, so I have minus 6x plus 8x, that gives us 2x, that gives us 2x, less than

09:27 equal to minus what, minus 50, now find the multiplicative inverse, remember the inverse

09:36 it will reverse out of the coefficient, so let's find the matrix inverse, so I'm going

09:41 to be having to ask what, 1 over 2, times 2x, less than equal to minus 50, minus 50, multiply

09:54 by 1 over 2, also, so this take out this, 3r1, 3r25, so I will have x less than equal to

10:05 what, minus 25, as my answer, so to check for my solution, what do I simply need to do,

10:12 I just simply need to pick this value, substitute it into the equation, then I can always check

10:19 for it, now, look at that, move to question number, question 3, say solve the 8 in equality

10:27 and check for your answer, so you could also follow to the process like I said, it's easy,

10:32 you just need to follow to the step by step, now if you look at it, collect the light

10:37 times together, that would be question number 3, collect the light times, what do I have,

10:42 I have 3x plus 5, greater than equal to minus 3 plus 4x, now I collect my light times together,

10:54 what do I have, I will have it as 3x, a half, minus 4x, remember, apply it to both sides,

11:00 then I have plus 5, a half, plus 5 minus 5, so greater than equal to minus 3, okay, plus 4x,

11:11 minus 4x, okay, I have plus 5, so that of the day, I am going to be having 3x minus 4x,

11:19 that would give me, that would give me what, I will have it as 3x minus 4x, 3x, sorry, minus

11:28 5x, minus x, this is x in between here, okay, so I will have it as what, minus x, then, okay,

11:41 minus x, moving up, so I will have, okay, greater than equal to, this would give me, taking

11:51 out the x, what do I, what am I supposed to, supposed to have, move up, 3 minus 5, minus

11:58 minus 5, so I will have it as what, minus what, 8, if you look to, minus x, minus x, that

12:08 give us minus 2x, greater than equal to, greater than equal to minus 8, so find the multiplicative

12:19 inverse of minus 8, so you just find the multiplicative inverse of minus 8, that would give us, one

12:25 of, minus 1 of 2 multiplied by minus 2x greater than equal to minus 8 multiplied by minus

12:32 1 of 2, so I have, this takes out this, minus takes out minus 3a, 1, 2a, 4, so I have x greater

12:41 than equal to minus 4, now, hold on, hold on, take it out, now, when you are dividing an

12:52 inequality sign, the sign changes, take note of it, when you are dividing an inequality

12:59 sign, with a negative, the sign changes, so when I am dividing, I am dividing through, remember,

13:06 sign the multiplicative inverse, I am saying, minus 1 of 2 multiplied by 2 minus 2x, this sign

13:15 automatically changes from greater than equal to, to less than equal to, minus 8 multiplied

13:23 by minus 1 of 2, because of the negativity to which you are dividing with, so that this

13:29 take out this, this take out this, will be less than equal to 2a, 1, 2a is 4, answer is 4.

13:38 Now, in an inequality, if you are dividing with a negative, to take note, the words you are

13:44 dividing with a negative, the inequality sign changes to the opposite, once you are dividing

13:51 with a negative, the inequality sign changes to the opposite, so you will see from the question

13:56 that, when I are dividing by negative, as it's okay, as a take note, this is negative, so because

14:02 the negative, the sign has to change from greater than equal to, to less than equal to, so if

14:08 it is greater than only, and I am dividing by negative, it's now going to change to less than

14:14 only, now, that's, because so easy, but take note, when you are dividing by negative,

14:22 the sign changes, okay, now, we will move on, we introduce a number line, like I said,

14:31 we could represent all what will be solving on a number line, I want a number line, the

14:36 number line is a straight line that divides equal segment and shows the relative position

14:42 of the numbers, that's a number line, so I will simply use your appropriate ruler to

14:52 appropriate ruler, I will bring out my ruler, okay, so I have this is my ruler, this makes

15:03 the job very easy, so you pick up this number line, you could pick up to start from, okay,

15:11 I'm adding it, yeah, okay, so, pick up like this, it will do, it has to be straight, it has

15:31 to be on a straight line, now, where I put it on a straight line, I could bring out my

15:35 ruler, just put it somewhere, yeah, and I would just say, yeah, it's 0, yeah, it's -2, yeah,

15:43 it's 2, for example, and once I do that, I can now represent it, you can see, so yeah,

15:50 it's negative, negative aspect, and yeah, it's the positive aspect of the number line, so

15:59 you could always use your number line to get, but ensure I use a straight ruler, a straight

16:06 ruler, which makes the job easy for you, so the position represents, now, if you will

16:12 say, this is greater than, greater than means is going this direction, you get it, greater

16:20 than equal to means is going this direction, but you are going to shade it, that's the

16:28 direction, but if it's going less than, less than means is going this direction, less than

16:36 equal to means is going this direction, but it has to be shaded, so you can see, this

16:42 science shaded, once they are going the direction, okay, now, let's move up, move up, move up the

16:50 scale, move up, move up, so, we could have a question number 1, solve the inequality and

16:57 show the solution on the number line, you could solve the inequality and show the solution

17:01 in the number line, so, you could straight up the process, and I have, okay, that's question

17:07 number 4, I want to show you the number line, so I have minus 1 minus 4x, greater than x plus

17:17 14, remember, using your additive inverse, everything goes with the additive inverse, I'll

17:24 have minus 1, minus 4x, remember, I want to go, I will have plus 1, because I want that

17:32 one, minus 1 to go, then I have minus x, greater than equal to, I will have x, I will have plus

17:39 14, I will have a, remember, I'm taking up, that's plus 1, I'm taking up the x, yeah, that's

17:47 minus x, so, I've applied it to both sides, so it goes, so 4x minus x, 4x, I will have

17:53 4x minus, minus 4x minus x, I will have minus 4x minus x left, greater than equal to, that's

18:01 greater than, this one goes, I will have 14 plus 1, so, 4x minus x, that gives us what, minus

18:10 what, minus 5s, greater than what, 15, now, I want to divide by negative, remember, if

18:18 I am dividing by negative, don't forget, the sign what changes, so, if I'm dividing

18:23 by negative, the sign changes, don't forget, so, once I do that, what does that mean, it means

18:28 that I change the sign changes to, I have minus 1 over 5, multiply by minus 5x, the sign changes,

18:35 remember, burn to the other side, becomes what, minus 1 over 5 times 15, so 5 here, 1, 5 here

18:44 is 3, 5 here, 1, 5 here, divide, I have x less than equal to minus 3, now, x less than

18:53 equal to minus 3, once I have x less than equal to minus 3, look at the sign to which it changes,

18:59 so, you just bring down your rule, your rulea, you get it, the point to which is going,

19:04 you just say, okay, I'm sticking up, yeah, yeah, you just choose, I'm okay, I'm picking

19:13 top here, from here, I'm taking up from here to, okay, okay, so I have it from here, straight

19:23 up, yeah, 4, while you put it that way, you could just bring down your rulea, and put it

19:30 as, come down and say, yeah, it's minus 3, bring it, take it up, suck it, what direction

19:38 is going, this direction, this direction, then you've gotten, you've solved your question,

19:44 you could see, simple, straight up, represent the number line, just follow your, your rulea

19:51 point, so once, follow your rulea point, or make sure the lines are straight, so if the lines

19:56 are straight, it becomes very, very what, easy, now, let's look at what problem involving

20:01 inequality, you could have what problem involving what inequality, it's a dual, enter the ways,

20:09 where it has to cycle and run, it has to cycle and ask to run, now, it cycles a distance

20:18 of 25 kilometer, and then run for 20 kilometer, cycles and run, now, it's average running speed,

20:27 it's half of it's average cycling speed, take note, now, Joe complete the ways in less

20:34 than 20 hour, what is his average speed for the ways, first, let the average running speed

20:42 be x, the average running speed, remember, the average cycling speed is twice the average

20:48 running speed, so, we have a question, question number 5, let the average speed, let the average

21:04 speed be x, so you have x, become the average speed, the average running, this is the terms

21:10 of the running, then in terms of the running is x, then in terms of the cycling, the cycling

21:21 is 2x, the cycling is 2x, okay, 2x, that's c, remember, for your formula to get your speed,

21:33 your speed is always distance divided by time, the speed is always distance divided by what,

21:40 time, that's your speed, so, what would not be your time, remember, if you now find your

21:46 time, which is t, would not be what, distance divided by speed, if you cos multiply, distance

21:52 divided by speed, so, find the distance for cycling, the c, distance that's used to cycle,

21:58 find the distance that's called, it does 25, over, what's the speed, 2, remember for cycling,

22:06 why for running, running is x, odd 9x, odd 9x, remember, so, the running, running we give

22:13 us 20 divided by x, okay, now bring them together, this is 25 divided by 2x, remember, because

22:24 we said it's twice, bring them together, bring them together, the two ways is not less than,

22:34 it's not less, it's not less than what, 12 hours, 30 minutes, so that is 25 over 2x plus 20 over

22:44 x, it's not less than 2 hours, 30 minutes, that's 2 times 1, 2 times 24 plus 1, 5, now I'll be 5 over 2,

22:51 bring it, now it has come into the form of what, into the form of a fraction, now with this, it becomes

23:00 easy, what do you do, find the LCD, what are the LCDs there, find the LCD, the I's, which is what, 2x,

23:07 LCDs, and that's what, 2x, so, use the multiplicative inverse for everything or 2, so I will have what,

23:16 2x multiply by 25 over 2x, okay, plus what, 2x multiply by 20 over x, I'll have 5 over 2 multiply

23:32 by 2x, get it, so this takes out this, successfully, x take out this, successfully, what do we have, we have

23:40 2 take out this, successfully, what's your finer answer, expect that your finer answer will give you what, 25,

23:49 okay, 25, plus what, plus 40, okay, less than what, 5 times x, now give us what, 5x, so that will give us 25 plus 40,

24:03 that gives us what, 65, 65, less than 5x, find your inverse 2, that's the multiplicative inverse,

24:13 down, multiply by 1 over 5, still multiply by 1 over 5, so we have 5, 5, 5, 1, 5 into this, we give us

24:22 what, 13, so we have 13, less than x, okay, so we have 13, less than x, so look at where it's going

24:33 to, okay, well, put on your number line, present the number line, you have this point, you move it up, down,

24:44 so done, move it, 13, 13, what direction is it facing, up, is this I could, no, no, this I could, just move it

24:55 this point, but so, it's average speed for running, it's 13, what would not be its cycling speed,

25:02 remember the cycling, the running is 13 already, then the cycling is twice the running, so that's

25:09 2 times 13, so that gives us also 26 of, that 26 kilometers, kilometers, so now with this two, we've been

25:23 able to represent on the number line, we've also been able to get this cycling speed and its

25:29 running speed, there is a need for you to understand how it works, inequalities on the number line and

25:35 inequalities are being represented with the greater than, less than equal to sign, and when does the

25:42 sign changes, so with this it gives you an idea, now we are moving to the point of evaluation,

25:49 evaluation says, the velocity V of a ball thrown directly up in the air is given by V equal to 20

26:02 minus 10 T, where T is the time in seconds, at what time will the velocity be between 10 seconds, 10

26:12 minutes seconds, and 15, solve the inequality and show the solution on the number line, so I have 5

26:19 per packet x plus 1, close the bracket, less than minus 15, solve an illustrated on the number line

26:28 like this, take note that the sign can change if it is division, okay, thank you very much for watching,

26:38 my name is Oyase Sunday, till next time.