Grade 10 Math | Solve Problems Quadratic Inequality in One Variable First Term (Term 1) Week 6

0:01 Hello everyone, welcome to native man mat tutorial wherein you can find free mathematics tagalog tutorial. Ang content ng video lesson na ito ay grade 10 mathematics under term 1 week 6 of the revise kicum. For today's video lesson, we are going to discuss solving word problems involving quadratic inequality in one variable. But before we discuss this lesson, if you are not yet subscribed to this channel, please subscribe for more video lesson. So what are you waiting for? Let's begin. The learning competency of this lesson is solve problems involving quadratic inequalities in one variable. Again, this is under term 1 week 6 of the

0:46 revise ken curriculum. And for the objectives at the end of the session you will be able to number one recall quadratic inequality in one variable number two identify the different inequality symbols and its mathematical phrase. Number determine the steps in solving problems involving quadratic inequality in one variable and number solve problems involving quadratic inequalities in one variable correctly. In solving problems involving quadratic inequality in one variable, we are going to follow these different steps. Step number one, read the problem carefully. Number two, identify inequality symbol

1:33 used. Number three, determine the given facts. Number four, translate into mathematical expression. Number five, solve the quadratic inequality. And number six analyze the results. Okay. For example number one or problem number one, Maya is instructed to make a rectangular flat with an area of atmos 48 s fo the length is 2 fet more than its wid. What are the possible measures of the width of the rectangular plat? Okay, so this problem is all about quadratic inequality in one variable. Now to solve this problem easily, we're

2:19 going to follow the different steps. Step number one, read and analyze the problem. So as you read analyze the problem also. Okay. So here the problem is all about making a rectangular flat and the area should be at most 48 s fo and the length is 2 fet more than its width. Okay. So here after analyzing the problem step number two, what is the inequality symbol used? So here the inequality symbol used from the given problem is less than or equal to. Why? Because the problem use the word atmost.

3:04 Okay? This one atmos means that is is less than or equal to in symbol. Okay. Then next after identifying the symbol, okay, identify the given data or facts. What are the given data? Okay, we have we have here 2 fet more than its width. So, we need the width, we need the length, and we need this one 48 s feet. Okay. So here let or W that is equal to W. Okay? And let length or L the length of the rectangular plat as wid

3:51 or w + 2. So bakit po W + 2 iyung length? From the given problem, the length is 2 ft more than its width. Okay meaning to say 2 fet is this one two and more than is plus and then that is we can be 2 + w or can be w 2 so pwede lang po yan okay so next is that the given atmos 48 s feet and what is this this is the area of the rectangle and the area a of the rectangle the formula is a is equal to length times width. Okay. So using this formula of area and then the given variable for the width and length,

4:38 we are going to create the so-called inequality of this problem. Okay? Again area is equal to the given fact area is equal to or area is less than or equal to 48 s foot. Okay. So from the given formula of the area, write the mathematical expression. So here the area is length times width its formula. So area is equal to length times width. Substituting the formula here, length times width for the area is less than or equal to 48 s foot. So from this inequality we can solve the length

5:24 or the width of the given rectangular plat. Okay. So how? Okay. So let's proceed in solving that in equality. Okay. So here substitute the given. So length times width. What is length of the given rectangular plat? So the length is w + 2. Okay, this one length is w + 2. Then times w is w for width is less or equal to 48 s fo we are going to simplify w + 2 w. So multiply this one using distributive property. So this will become what is w + w s and then w 2 is 2w. Okay? So we

6:15 have now plus here. So w s + 2w is less or equ to 48. So this is now the inequality that we are going to use to solve what is asked or what is being asked in the problem. Okay, after deriving the inequality of the given problem then we are going to solve this inequality to answer what is being asked in the problem. So again the inequality is w s + 2w is less or equ to 48. So for step number one we are going to what? Write the inequality in standard form. So what is the standard form of quadratic inequality?

7:02 The right of inequality symbol should be zero. But the number there is 48. So we move this 48 to the left of this inequality. So this will become w s + 2w. So this 48 if you move to the left sign uh left side of the inequality by applying addition property of equality then that will become -48. So 48 that will become -48. So that's why - 48 so is less than or equal to zer. So the right of inequality is zer. Now after this we are going to write this inequality into equation and solve the equation using any method. So notice here in our inequality.

7:48 So the given is that factorable. So we're going to factor the inequality or equation. So that's why the symbol here is equal sign. Okay. So is it correct that the factors are w- 6 from this inequality or equation? Okay. So let's check. 8 * -6 is -48. That's correct. And then add or subtract from the middle term. What is 8- 6? Is it 2? Yes, that's correct. So here the factors of this equation or inequality is 4 + uh w + 8 * w - 6 and then get the

8:34 value of the w here of the variable. So we have here + 8 so the value of w is -8. So here -6 so the value of w is + 6. Now after determining the two values of W or this is called roots of the quadratic equation now we are going to move on step number four we are going to plot this two value of W in the number line. Okay, we have here first value is -8 this one and the second one is + 6. Okay, this one. So, we are going to plot two points there. So, from the two points on the number line, it divides

9:19 the number line into three parts. So, the first part is that from the point -8 going to the left or going to negative infinity. So the inequality notation is W is less or equ to -8. How about the middle part from -8 to6? So the inequality there is -8 is less or equ to W is less or equal to 6. Okay, this one from -8 to+ 6. And how about the third part? From pos 6 to going to the positive infinity. So here the inequality is W is greater than or equal to 6. So from the three parts or from

10:07 the three inequality notation we are going to choose one point to check if the given part or inequality is part of the solution or not. Okay? From the first part W is less or equ to -8. I want to choose or [sumisingasing] I choose - 10. Is 10 part of this inequality? Yes, because the inequality says that from -8 going to the negative infinity. So -8 -10 -8 -9 -10-1 and so on. So I choose -10. How about from -8 to 6? So I choose zer there to make it easy. So W is 0. And for the third part,

10:57 okay, greater than or equal to 6. So do not choose 6 because that is the critical point. So you can choose 7 going to pos infinity. So I choose there. Okay. W 10. I choose pos 10 here. Okay, for step number 5, we are going to test if the given value is a solution or not a solution of the given inequality. So again, we choose - 10 from the first inequality notation. We choose from the middle and we choose pos 10 for the third inequality notation. Okay. Again, what are the factors of the given

11:42 inequality or equation? W + 8 and W - 6. So, here is the inequality. W + 8 * W - 6 is less or equ to 0. Okay. And we set that inequality as negative. If the result is negative then that is true. But if the result is positive that is not true or that is false. Okay we test for the first part - 10. So we substitute we substitute this 10 to the given factor and uh uh get the sign of the result if positive or negative. Okay. So -10. So

12:28 -10 is the W. So -10 + 8 is -2. So the sign of the result is negative. So we're going to write here negative. How about the second factor? -10 again. So -10 - 6 that is -1. So the sign of the result is negative. So we write here negative. Okay? So if you're going to simplify negative negative is positive. Now we set as negative to become true the result is positive then the result is false. Okay so positive is false. How about W is 0? So the value is 0. So we are going to substitute here. So 0 + 8 is8.

13:15 So mean to say the sign there is positive. How about for the second factor 0? So 0- 6 is -6. So the sign of the result or the answer is negative. So we write there negative and then multiply positive times negative is negative. Mean to say since negative is the result then is true. Okay. How about the last part? So we choose 10 substitute here. 10 + 8 is 18. So pos 18. So we write positive there. Then the next factor w - 6. So 10 10 - 6 is + 4. So the result is + 4. Okay. So here

14:02 positive and then multiply positive times positive that is equal to positive. Again is positive less than or equal to zero? No. Okay. So that is false. Okay. So here ang nag-negative lang po ay iyung middle part. mean to say - 8 is less than or equal to w is less than or equal to 6 is the notation that we are going to choose because it is true using signed test. Okay. For step number six, we are going to choose what are the possible measures of the width of the rectangular plant. Okay. So ito po yung nagtina wherein -8 is less

14:48 or equ is less than or equ to 6. So what does it mean with this inequality? Ito lang po yun. Ang value daw ng ng W or with here is from -8 to6. Kaya nilista ko po lahat para hindi po kayo malito. So we have -7 ah -8 -7 -6 up to6. Now here in choosing the wid or the length of the wid we need to think that for negative measurement there is no negative. Is that clear? So wala pong negative na measurement. So ang chu-choose lang po natin diyan ay positive length. Okay? Hindi rin po pwede yung zero kasi kapag meron kang

15:35 rectangular flat ibig sabihin hindi po zero ang kanyang wid. Meron at meron po siyang wid. So ichu-choose lang po natin 'yung positive value. Again, there is no negative value. So we do not choose negative result here. Okay? So that means that the value of the width or the length of the width starts from pos 1 up to pos 6. Okay. To answer in sentence, the possible measures of width of the rectangular plat are 1 m. Okay, this 1, 2 m, 3 m, 4 m, 5 m and 6 m.

16:21 Okay. So, how about this problem number two or example number two, the length of a rectangular garden is 6 m twice its width. Find all possible measure of the widular garden not exceeding 80 s met. Okay. So here from the given problem to follow the steps to solve that one. Okay. First step is that read and analyze the problem. So after reading or as you read please analyze what is there in the problem. Okay. So the problem is all about rectangular garden. It says there that 6 met or the length should be

17:07 6 m twice its width. Okay. So here the result in the rectangular garden should not exceed 80 s me. Okay. So here for step number two, what is the inequality symbol used? So ano po yung word na ginamit to derive this inequality symbol? So we have the word not exceeding 80 s me. So from the word not exceeding ibig sabihin hindi po pwedeng lumampas ng 80 s m. So that's why we use is less than or equal to. Okay? And then for step number three,

17:52 what are the given facts? So ito po yung given facts. Hindi po given yung with. Nakalagay lang po dito with. Okay? So we let with W as with or W. Then we have this one. We we let length L as what is the length of the given rectangular garden? The length is 6 m more than twice its wid. Okay? So here 6 m more than is plus. Twice is 2 then W is wid. Can be 6 + 2w or pwede naman pong 2w + 6. So that is the length of the given rectangular garden. And again the formula of the area of the rectangle is

18:39 length times width. Okay? We're going to substitute the given from this formula. So area is equal to or area is less than or equal to 80 s feet. Okay? Yun po nakalagay dito or 80 squ rather. Okay. So, 80 squ meters. So, we're going to substitute the given there to derive the mathematical expression or the mathematical inequality. So this will become area from the formula is length times width. So we have here length times width. Then copy is less than or equal to 80. Okay. Substituting the value of length here because length is

19:24 2w+ 6. So this will become 2w + 6 for length then copy times w is w there is less than or equal to 80. And then next multiply again this one binomial by a monomial by distributive property. So this will become w 2w is 2w s then plus w 6 is 6w. then copy is less than or equal to 80. So this is now the inequality that we're going to use to solve what is being asked in the problem. Okay. After deriving the inequality from the problem then we are going to solve

20:11 this inequality to answer what is being asked here in the problem. Okay? For step number one, we're going to write an equality into standard form. So 'yung 80 i-move natin sa left side. Dapat ang right of the inequality is zero. So this become 2w² + 6w - 80 is less or equ 0. So from pos 80 that will become -80 kaya po - 80 here. Then next notice here yung atin pong numbers 2 6 and 80 ay meron po siyang common factors. Meaning to say we can simplify this in equality para mas madali po nating i-solve. So lahat po ng numbers ay divisible by 2. So we are

20:57 going to divide 2 to all the terms. Okay? So here 2/ 2 is 1. Okay? So this become 1w s or just W s + 6w / 2. So 6 / 2 is 3. And then yung W kaya 3W. And then we have minus 80 / 2 is 40. And then copy is less than or equal to 0. So ito medyo nasimple na po yung ating inequality and from inequality move to equation and from that equation we are going to find the roots because that is quadratic equation. Okay? So by calculating the roots what method that we're going to use? Can we use factoring or quadratic formula? So kapag hindi po

21:45 factorable itong given then quadratic formula. But the given is factorable. And what are the factors? So here W + 8 W - 5. So how? Bakit po? If you're going to multiply 8 -5 because that is -5. 8 -5 is -40. So that's why- 40 here. How about add or subtract? 8- 5 is 3. So we have here 3W. So the factors are correct. So mula po dito we are going to find the value of w. So here the value for the first factor is8 because that is + 8. And then the value of the second factor since -5

22:32 kaya po 5. So from the given roots of this uh equation we are going to plot that on the number line. Okay? Step number four i-flat natin sa number line. So -8. Okay, ito po yung -8. The second root is pos 5. Okay, this one. Okay, so after platting the two numbers or roots on the number line, we are going to write the inequality notation. For the first part, from -8 going to the left. So, we have here W is less than or equal to -8. Mean to say the value the value of W is from -8 going to the left. or

23:17 going to negative infinity and for the second part from -8 to 5. So we're going to write its inequality -8 is less or equ is less than or equal to 5. And for the last part from pos 6 going to the right or going to positive infinity. So we have here W is greater than or equal to 5. Okay. After that we are going to choose one number or one integer from the given parts or inequality. So ito pong from -8 to infinity I choose there -9. Okay? -9 is part of this inequality. How about for the second part from -8 to 5? So

24:06 kasama yung zero. So I choose zer here. And for the third part, W is greater than or equal to 5. So I choose pos 6. Okay? So mula po dito sa different values, we are going to test if that value is a solution or not a solution. Again using the sign test we can test if the inequality notation is a solution or not a solution of the given inequality. Okay. So here again -9 0 and 6 and the factors are here w + 8 and w - 5. Okay here again w + 8 * w - 5 is less or equ 0. So ibig

24:55 sabihin is less than or equal to zer is-set natin na negative. Kapag negative yung result then true. Kapag positive yung yung result then false. Okay? So i-test natin yung -9. So -9 sa first factor -9 + 8 is -1. So the sign of the answer is -1. So negative here. Next -9 - 5 is -14. So the sign is negative. Okay? If you're going to multiply negative times negative positive mean to say positive yung result so hindi pupwede. So that is false. How about sa second inequality notation? 0. So 0 + 8

25:42 is 8. So 8 next 0 - 5 is -5. Okay? So -5 that's why negative. If you're going to multiply positive times negative, the answer is negative. Since the result is negative then that is true. Mean to say ang true po ay itong middle. How about the last part? Check natin if true or false. So pos 6. 6 + 8 is 14. So 14 so that's why positive here. Next 6 - 5 is pos 1. So again positive. So what is positive times positive? That is positive. To say positive po yung result

26:27 that is false. Dapat po ang result ay negative. Okay? So the inequality that make it true using sign test is that the middle inequality. So -8 is less than or equ is less or equ to 5. Okay. From the given inequality that make true, we are going to choose the value of the wid. Again, hindi po pwedeng maging value yung negative. So from the given inequality, i-list po natin lahat from -8 to 5. So -8 -7 until 5. And then again we are going to choose the positive value. So to say the possible values of the width of the

27:15 rectangular garden are 1 2 3 4 and 5. Okay. So we're going to write it here. The wid are 1 2 3 4 and 5. Okay. So to answer the question, the possible measures of width of the rectangular garden are 1 m, 2 m, 3 m, 4 m and 5 m. Okay, another example or problem. Number 3. The length of a rectangular study table is 3 fet more than twice its width. Find all possible wids if the table area must be at least 54 s feet. Okay. So this is the problem. Again this

28:03 problem is all about quadratic inequality. So to solve we're going to follow the steps. So read and analyze the problem. So what is the problem all about? So this is about the rectangular study table. So it says there the length is 3 fet more than twice its wid. Okay? And the area must be at least 54 s feet. Okay? So yun po yung pagkakaintindi po natin. Okay. Next is that we are going to identify the inequality symbol. So here it the word at least. So what does it mean with at least in symbol is greater

28:48 than or equal to? 'Yun po yung ibig sabihin ng at least. And then next identify the given fact. So here we are going to set that with W is W. And then how about length? So the length should be from here the length of a rectangular table is 3 fet more than twice its width. So 3 + twice is 2 and with so 3 + 2w. Okay, this one 3 + 2w. So can be written in 2w + 3. Pareho lang po yun. 'yung 3 + w 'yung 3 + 2w and 2w + 3 are just the same. Okay? So again from the

29:35 formula of the area of the rectangle because the given figure here is a rectangle so area is equal to length times width. So here we are going to identify the area of the given rectangular study table. So the area is the symbol is greater than or equal to 54 s feet. Okay. So here from the given inequality substitute what is there in our given fax. Okay. So the area is length times width. Again length times width and then copy is greater than or equal to 54 s feet.

30:21 Okay. So next substitute what is the length of the given problem? So the length is 2w + 3. So substitute here 2w + 3 W. What is W? W there is greater than or equal to 54. Okay. Next. By substitution or by distributive property should be okay. This will become W * 2w is 2w S. And then W * 3 is 3W. Okay. So here copy is greater than or equal to 54. Okay. So that inequality is the inequality that we are going to use to solve what is asked in the problem. Okay.

31:10 So after deriving the inequality now we are going to solve this one. So again we are going to write the inequality in standard form. The right side of the inequality should be zero. So we have 54 here. So we move 54 to the left of inequality. So this will become 2w² + 3w - 54 is greater than or equ to 0. Okay. So + 54 that will become - 54 here and then change from inequality to equation. Okay? And then we're going to solve this equation by factoring or quadratic formula. So ito bang equation ay factorable or not factorable? So that is

31:56 factorable. So we are going to factor it. So the factors are 2w - 9 * w + 6. Okay? From the given factors, we are going to write its roots. Okay? So yung roots po ng first factor 2w - 9 is 9 / 2. So paano po nakuha yung 9 / 2? This will become 9 divided by the numerical coefficient of W2. So 9 / 2. Okay, this one and this is minus. So the root is positive because that is minus. So here + 6 or + 6 that will become -6. So the root here is -6. So we have this two values or two roots

32:41 of the equation then we are going to plot that on the number line. So we have here -6. Okay this 1 -6. How about 9 / 2? So in decimal 9 / 2 is 4.5. Okay? Mayro'n po tayong 4.5. Ito po 'yan. Okay. So, we're going to plot it on the number line. So, -6 this one and 9/2 or 4.5 is this one. Okay. So again, the two points divide the number line onto into three parts. So, ito po yung first part again from -6 to infinity. So, W is less or equ6 and the middle part is from6

33:27 to to 4.5. So this become -6 is less or equ to W is less or equ to 9 / 2 or 4.5. And for the third part from 4.5 or 9 /2 going to the positive infinity. Okay. Now we're going to choose points from this three parts of this number line. Okay. So here I choose for the first part. So from -6 going to the negative infinity so part is -7 here ang dito po sa gitna kasama po yung zero. So I choose zer and here ang greater than 9/2 po ay 5. So 5 is part

34:14 of the inequality here. Okay. So next for step number 5 we are going to test if the given value we choose is a solution or not a solution. Okay. So here the factors are w - 9 / 2 or 4.5 and W + 6. So if you're going to set it in inequality, so the two factors is greater than or equal to 0. So here dapat ang na lalabas ay positive because if positive positive is greater than or equal to 0. So kapag ang result is negative false if the result is positive and that is true. Okay? So here start with - 7 -7-

35:05 9 / 2. So negative and then minus that will become negative. So the result is negative there. How about second factor? -7 + 6 is -1 then the sign is negative. and multiply negative is positive. Since the result is positive then that is true okay how about second 0 so 0 - 9 / 2 or 0 - 4.5 that is -4.5 5 or 9. So that's why negative here. How about 0 + 6 is 6 or pos 6? So the sign is positive. So here positive. So what is negative times

35:51 positive? That is negative. Since the result is negative then that is false. So sinet po natin dapat ang result ay positive para true. If negative then that is false. How about the third one? 5. 5 - 9/2 or 5 - 4.5. So ang lalabas po diyan since mas malaki po yung 5 sa 4.5 then the result is positive. How about second factor? 5 + 6 is 11 or + 11. So that is positive and then multiply positive times positive is positive. Since the result is positive then our conclusion is true. Okay? So here that is true. So in our test there are two

36:39 sets of inequality which is true. This one the first one and the third part. Okay? So next step number six. Ito po yung true kanina sa ating test. W is less -6 mean you say from -6 to infinity. And take note there is no negative measurement. So hindi po pwede ang -6 to infinity. Okay? Kahit na po nag-tring inequality but we choose only the positive number. Okay, here W is greater than or equal to 9/2. So since that is positive then we choose that one as the possible wid. So this is our

37:26 answer. Since the wid wids if the table's area must be at least 54 s feet are greater than or equal to 9/2 fet or 4.5 fet. So mula po 9/2 going to positive infinity. Okay? So yun po yung possible wids of the rectangular study table. Okay? So let's see your understanding if you really understand about solving problems involving quadratic inequality. So answer all the question being asked in the problem. This is the problem. Okay. A resort is constructing a square

38:13 pool. The area of the pool must be less than 400 s me. Okay. So here the problem is all about constructing square pool. So this is or these are the questions. Question number one, what is the given data? So ano po yung given data natin? This is the answer. 400 s me. Okay, this one. How about question number two? Write the inequality that describes the problem. So I will give you some seconds or minutes to answer this question number two or pwede po ninyong i-post yung video na ito. Try to answer first and then later i-play po ninyo and check if your answer is correct. Okay? So sige

38:58 i-reveal po natin question number two. Write the inequality that describes the problem. So the inequality is this one. X is less 400. And for question number 3, what are the possible sid lens of the pool? Okay, again I will give you uh 30 seconds to 1 minute to answer this one. And you can post this video kung kulang po yung 30 seconds and then play it again later. So the possible sideles of the full are okay this one ang inequality po natin ay 0 is less x is less 20 then

39:44 this will become the side length of the square pool can be any positive value less than 20 m. Okay. So kahit na ano pong value less than 20 meters but hindi po aabot ng zero kasi wala pong length na zero. Okay to end up our discussion try to do this activity number two, problem solving. So this problem ay iiwan ko po sa inyo as your activity. Try to read, analyze and solve what is being asked. And then later after solving, try to write or comment your answer in the comment section below. Okay,

40:31 thanks for watching guys. I hope that you learned from my simple video lesson. If you like this video, please click like and subscribe if you are not yet subscribed to this channel for more video lesson. y