Solving Inequalities Using Multiplication or Division - Free Educational videos for Students in K-12

Solving Inequalities Using Multiplication or Division - By Anywhere Math

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00:0-1	Welcome anywhere . Math . I'm Jeff Jenkinson and today
00:02	we're solving inequalities using multiplication or division . However ,
00:08	there will be no negative numbers in this video .
00:12	So with that let's get started . Okay , example
00:32	one solve X over five is less than or equal
00:35	to two and then grab the solution . Well ,
00:39	solving inequalities using multiplication or division is just like solving
00:44	equations with multiplication or division . As long as you're
00:49	not multiplying or dividing negative numbers . And in this
00:52	video it's all going to be positive . So it's
00:55	gonna be like the same . It's going to be
00:57	just like doing it with equations . Our goal is
01:00	to get the variable alone right now . X is
01:02	being divided by five . So I do that inverse
01:05	operation and multiply this side by five to get that
01:09	X alone and anything I do to one side ,
01:11	I have to do the same to the other side
01:14	to make sure the inequality stays true . So here
01:17	we go . Uh X over five . If I
01:19	multiply this side by five Less than or equal to
01:22	. To do the same thing on this side multiply
01:25	that side by five . Those fives will cancel out
01:29	. And I'm left with X is less than or
01:31	equal to 10 . Okay , I've solved it now
01:35	. Let's grab that solution . Here is my quick
01:40	Number line . I know x needs to be less
01:43	than or equal to 10 . So maybe here is
01:45	zero , there is 10 less than or equal to
01:50	because it's or equal to . That means my dot
01:53	needs to be like that filled in Arrow going towards
02:00	the less than theirs . Example one here's some to
02:03	try on your own . Okay , example to solve
02:13	for N is greater than 32 and then grab the
02:16	solution . So same thing . I'm trying to get
02:19	the variable alone . I notice and is being multiplied
02:22	by four . So the inverse operation of multiplication is
02:26	division divided by four . Well undo that multiplication and
02:30	then I do the same thing here . So four
02:33	end is greater than 32 . I'm going to divide
02:36	this side by four , which means I'm also gonna
02:39	divide that side by four to make sure the inequality
02:43	stays true . Uh those get cancelled out and I
02:46	get N is greater than 32 , divided by four
02:50	, is eight . Okay , um This represents all
02:54	my solutions to this inequality . Any number that's greater
02:57	than eight would be a solution to this inequality .
03:01	So let's graph that here is a quick little number
03:06	line . I'm gonna be at eight , let's say
03:09	eight is here let's go 89 10 , something like
03:14	that at eight because it's greater than it's not greater
03:18	than or equal to . I'm going to have an
03:20	open circle and greater than which means the arrows going
03:24	to the right . So again , This arrow represents
03:29	all the solutions to this inequality . Any number that's
03:34	greater than eight would be a solution , right ?
03:38	10 would be a solution because four times 10 is
03:40	40 that's greater than 30 to 9 would be a
03:43	solution . Four times nine is 36 that's greater than
03:46	32 . A million would be a solution . Anything
03:49	greater than eight . Let's try one more . Example
03:52	, example three A one day pass to a gym
03:55	cost $4.50 a 30 day pass for like a month
03:58	long pass cost $45 . So part a right And
04:02	solving inequality to find when it's better to buy the
04:06	one day passes instead of buying the full month long
04:10	, 30 day pass . So this is very typical
04:13	right ? New Year's resolutions . People say , okay
04:15	, I want to go to the gym . Well
04:17	, is it better to buy the month long pass
04:19	or is it better to buy or to pay each
04:22	time you go ? Well , it depends on how
04:24	often you go , which is exactly what we're trying
04:26	to find out here . So uh let's look ,
04:30	a one day pass costs $4.50 . The month long
04:35	is $45 . Now , the one thing we don't
04:39	know , it's how often we're gonna go , how
04:42	many days we're going to go ? That's gonna be
04:44	our variable , right ? Whenever you're writing inequalities or
04:48	equations , always think what's your variable gonna be ,
04:51	what , don't , you know ? And in this
04:52	case we don't know how often how many days we're
04:56	going to go to the gym . So that's gonna
04:57	be my variable . I'm gonna call that d the
05:00	for days Uh to the gym . Well , each
05:03	day costs $4.50 . So this will become 4.5 d
05:11	. Right ? $4.54 . 50 cents times the number
05:14	of days you go now , what we're looking for
05:18	is when it's going to be cheaper , right ?
05:22	When it's better to do the one day passes and
05:26	it would be better when this cost is cheaper than
05:32	The $45 than the month long . So to show
05:35	that this is cheaper or a better deal , we
05:37	would make it less than 45 . So take a
05:44	second and just look at that and think about what
05:46	we what we wrote . So what we're saying is
05:49	$4.50 times the number of days , right ? Is
05:56	less than if that's less than $45 , which is
06:00	the month on pass . This would be the better
06:03	deal . So now let's solve it . So we
06:05	wrote the inequality now let's solve it . Uh divide
06:09	both sides by 4.5 . To get that d .
06:12	That variable alone . D . Is less than ,
06:18	well 45 divided by 4.5 is 10 . You can
06:24	do that over here or just trust me . Uh
06:28	What that means is when is it better to buy
06:31	a one day pass ? Well it's better to buy
06:33	a one day pass if You go less than 10
06:38	days . So there is our answer . Let's try
06:41	one more . Okay . Same problem . We're just
06:43	gonna do part B . And part C . When
06:46	would the cost be equal ? The cost would be
06:48	equal if you went exactly 10 days , Right ?
06:53	If you in 10 days 10 times $4.50 would be
06:57	$45 . If you went 10 days with a month
07:00	passed . Well still $45 and last one . When
07:03	would it be better to buy the 30 day pass
07:06	? Well , if it was cheaper To buy the
07:10	day pass , if you went less than 10 days
07:14	And if you do exactly 10 days , it doesn't
07:16	matter . Well then if you go more than 10
07:19	days , the 30 day pass would be the better
07:22	deal . Because after every day above 10 , it
07:26	doesn't matter . You don't get charged anymore because it's
07:29	just a 45 straight $45 a straight fee . No
07:33	matter if you go 11 , 12 , 13 ,
07:36	if you go 25 days or if you go all
07:38	30 right , that would be the better deal .
07:41	Because if we did 30 days and pay $4.50 each
07:44	day . Well , 30 times 45 is gonna be
07:47	much greater than 45 days . Okay ? Here are
07:51	some to find your own as always . Thank you
08:01	so much for watching . And if you like this
08:03	video , please subscribe .

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