[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.99,0:00:04.06,Default,,0000,0000,0000,,The expression 5X
Dialogue: 0,0:00:04.06,0:00:10.97,Default,,0000,0000,0000,,minus 4. Greater than\Ntwo X plus 3 looks like an
Dialogue: 0,0:00:10.97,0:00:15.67,Default,,0000,0000,0000,,equation, but with the equal\Nsign replaced by an Arrowhead.
Dialogue: 0,0:00:16.99,0:00:19.14,Default,,0000,0000,0000,,This denotes that the.
Dialogue: 0,0:00:19.72,0:00:24.76,Default,,0000,0000,0000,,Part on the left, 5X minus four\Nis greater than the part on the
Dialogue: 0,0:00:24.76,0:00:26.20,Default,,0000,0000,0000,,right 2X plus 3.
Dialogue: 0,0:00:27.44,0:00:29.91,Default,,0000,0000,0000,,We use four symbols to denote in
Dialogue: 0,0:00:29.91,0:00:34.36,Default,,0000,0000,0000,,Equalities. This symbol\Nmeans is greater than.
Dialogue: 0,0:00:36.84,0:00:41.04,Default,,0000,0000,0000,,This symbol means is greater\Nthan or equal to.
Dialogue: 0,0:00:42.14,0:00:44.97,Default,,0000,0000,0000,,This symbol means is less than.
Dialogue: 0,0:00:45.89,0:00:50.18,Default,,0000,0000,0000,,On this symbol means is less\Nthan or equal to.
Dialogue: 0,0:00:51.38,0:00:55.52,Default,,0000,0000,0000,,Notice that the Arrowhead\Nalways points to the
Dialogue: 0,0:00:55.52,0:00:56.55,Default,,0000,0000,0000,,smaller expression.
Dialogue: 0,0:00:58.78,0:01:01.58,Default,,0000,0000,0000,,In Equalities can be\Nmanipulated like equations
Dialogue: 0,0:01:01.58,0:01:03.58,Default,,0000,0000,0000,,and follow very similar\Nrules.
Dialogue: 0,0:01:04.94,0:01:06.100,Default,,0000,0000,0000,,But there is one\Nimportant exception.
Dialogue: 0,0:01:08.79,0:01:13.92,Default,,0000,0000,0000,,If you add the same number to\Nboth sides of an inequality, the
Dialogue: 0,0:01:13.92,0:01:17.88,Default,,0000,0000,0000,,inequality remains true. If you\Nsubtract the same number from
Dialogue: 0,0:01:17.88,0:01:22.62,Default,,0000,0000,0000,,both sides of the inequality, it\Nremains true. If you multiply or
Dialogue: 0,0:01:22.62,0:01:26.56,Default,,0000,0000,0000,,divide both sides of an\Ninequality by the same positive
Dialogue: 0,0:01:26.56,0:01:28.14,Default,,0000,0000,0000,,number, it remains true.
Dialogue: 0,0:01:29.86,0:01:33.96,Default,,0000,0000,0000,,But if you multiply or divide\Nboth sides of an inequality by a
Dialogue: 0,0:01:33.96,0:01:36.00,Default,,0000,0000,0000,,negative number. It's no longer
Dialogue: 0,0:01:36.00,0:01:40.22,Default,,0000,0000,0000,,true. In fact, the inequality\Nbecomes reversed. This is quite
Dialogue: 0,0:01:40.22,0:01:45.07,Default,,0000,0000,0000,,easy to see because we can write\Nthat four is greater than two.
Dialogue: 0,0:01:45.86,0:01:50.80,Default,,0000,0000,0000,,But if we multiply both sides of\Nthis inequality by minus one, we
Dialogue: 0,0:01:50.80,0:01:51.94,Default,,0000,0000,0000,,get minus 4.
Dialogue: 0,0:01:52.44,0:01:54.99,Default,,0000,0000,0000,,Is less than minus 2?
Dialogue: 0,0:01:55.71,0:01:57.79,Default,,0000,0000,0000,,We have to reverse the\Ninequality.
Dialogue: 0,0:01:58.92,0:02:05.34,Default,,0000,0000,0000,,This leads to difficulties\Nwhen dealing with variables
Dialogue: 0,0:02:05.34,0:02:11.75,Default,,0000,0000,0000,,because of variable can\Nbe either positive or
Dialogue: 0,0:02:11.75,0:02:14.70,Default,,0000,0000,0000,,negative. Look at these two
Dialogue: 0,0:02:14.70,0:02:17.24,Default,,0000,0000,0000,,inequalities. X is greater than
Dialogue: 0,0:02:17.24,0:02:19.71,Default,,0000,0000,0000,,one. And X squared.
Dialogue: 0,0:02:20.23,0:02:21.65,Default,,0000,0000,0000,,Is greater than X.
Dialogue: 0,0:02:23.38,0:02:27.57,Default,,0000,0000,0000,,Now clearly if X squared is\Ngreater than ex, ex can't be 0.
Dialogue: 0,0:02:28.28,0:02:31.88,Default,,0000,0000,0000,,So it looks as if we ought to be\Nable to divide both sides of
Dialogue: 0,0:02:31.88,0:02:34.23,Default,,0000,0000,0000,,this inequality by X. Giving us.
Dialogue: 0,0:02:34.74,0:02:38.09,Default,,0000,0000,0000,,X greater than one, which is\Nwhat we've got on the left.
Dialogue: 0,0:02:39.75,0:02:43.13,Default,,0000,0000,0000,,But in fact we can't do this.\NThese two inequalities are not
Dialogue: 0,0:02:43.13,0:02:47.10,Default,,0000,0000,0000,,the same. This is because X\Ncan be negative.
Dialogue: 0,0:02:48.44,0:02:53.06,Default,,0000,0000,0000,,Here we're saying that X is\Ngreater than one, so X must be
Dialogue: 0,0:02:53.06,0:02:56.28,Default,,0000,0000,0000,,positive. But here we have to\Ntake into account the
Dialogue: 0,0:02:56.28,0:02:57.58,Default,,0000,0000,0000,,possibility that X is negative.
Dialogue: 0,0:02:58.18,0:03:04.60,Default,,0000,0000,0000,,In fact, the complete solution\Nfor this is X is greater than
Dialogue: 0,0:03:04.60,0:03:07.81,Default,,0000,0000,0000,,one or X less than 0.
Dialogue: 0,0:03:08.42,0:03:11.45,Default,,0000,0000,0000,,Because obviously if X is\Nnegative, then X squared is
Dialogue: 0,0:03:11.45,0:03:15.39,Default,,0000,0000,0000,,always going to be greater than\NX. I'll show you exactly how to
Dialogue: 0,0:03:15.39,0:03:18.42,Default,,0000,0000,0000,,get the solution for this type\Nof inequality later on.
Dialogue: 0,0:03:20.93,0:03:23.93,Default,,0000,0000,0000,,Great care really has to be\Ntaken when solving inequalities
Dialogue: 0,0:03:23.93,0:03:27.53,Default,,0000,0000,0000,,to make sure that you don't\Nmultiply or divide by a negative
Dialogue: 0,0:03:27.53,0:03:33.16,Default,,0000,0000,0000,,number by accident. For example,\Nsaying that X is greater than Y.
Dialogue: 0,0:03:34.14,0:03:40.95,Default,,0000,0000,0000,,Implies. That X\Nsquared is greater than Y
Dialogue: 0,0:03:40.95,0:03:44.06,Default,,0000,0000,0000,,squared only if X&Y are
Dialogue: 0,0:03:44.06,0:03:51.13,Default,,0000,0000,0000,,positive. I'll start\Nwith a very simple
Dialogue: 0,0:03:51.13,0:03:57.96,Default,,0000,0000,0000,,inequality. X +3 is\Ngreater than two.
Dialogue: 0,0:03:59.04,0:04:03.05,Default,,0000,0000,0000,,To solve this, we simply need\Nto subtract 3 from both sides.
Dialogue: 0,0:04:03.05,0:04:07.72,Default,,0000,0000,0000,,If we subtract 3 from the left\Nhand side were left with X. If
Dialogue: 0,0:04:07.72,0:04:11.73,Default,,0000,0000,0000,,we subtract 3 from the right\Nhand side were left with minus
Dialogue: 0,0:04:11.73,0:04:14.74,Default,,0000,0000,0000,,one and that is the solution\Nto the inequality.
Dialogue: 0,0:04:15.93,0:04:19.27,Default,,0000,0000,0000,,In Equalities can be represented\Non the number line.
Dialogue: 0,0:04:21.32,0:04:25.34,Default,,0000,0000,0000,,Here are solution is X is\Ngreater than minus one.
Dialogue: 0,0:04:26.24,0:04:28.48,Default,,0000,0000,0000,,So we start at minus one.
Dialogue: 0,0:04:30.28,0:04:32.67,Default,,0000,0000,0000,,And this line shows the range of
Dialogue: 0,0:04:32.67,0:04:35.11,Default,,0000,0000,0000,,values. The decks can take.
Dialogue: 0,0:04:36.30,0:04:40.15,Default,,0000,0000,0000,,I'm going to put an open circle\Nthere. That open circle denotes
Dialogue: 0,0:04:40.15,0:04:42.08,Default,,0000,0000,0000,,that although the line goes to
Dialogue: 0,0:04:42.08,0:04:46.64,Default,,0000,0000,0000,,minus one. X cannot actually\Nequal minus. 1X has to be
Dialogue: 0,0:04:46.64,0:04:47.97,Default,,0000,0000,0000,,greater than minus one.
Dialogue: 0,0:04:49.20,0:04:55.40,Default,,0000,0000,0000,,Let's have a\Nlook at another
Dialogue: 0,0:04:55.40,0:04:56.44,Default,,0000,0000,0000,,one.
Dialogue: 0,0:04:58.44,0:05:01.43,Default,,0000,0000,0000,,4X plus 6.
Dialogue: 0,0:05:02.06,0:05:05.80,Default,,0000,0000,0000,,Is greater than 3X plus 7.
Dialogue: 0,0:05:07.21,0:05:12.31,Default,,0000,0000,0000,,First of all, I'm going to\Nsubtract 6 from both sides, so
Dialogue: 0,0:05:12.31,0:05:16.98,Default,,0000,0000,0000,,we get 4X on the left, greater\Nthan 3X plus one.
Dialogue: 0,0:05:17.92,0:05:22.54,Default,,0000,0000,0000,,And now I'm going to subtract 3\NX from both sides, which gives
Dialogue: 0,0:05:22.54,0:05:24.31,Default,,0000,0000,0000,,me X greater than one.
Dialogue: 0,0:05:25.08,0:05:29.04,Default,,0000,0000,0000,,And again, I can represent this\Non the number line.
Dialogue: 0,0:05:29.78,0:05:31.81,Default,,0000,0000,0000,,X has to be greater than one.
Dialogue: 0,0:05:33.68,0:05:35.40,Default,,0000,0000,0000,,But X cannot equal 1.
Dialogue: 0,0:05:36.99,0:05:43.77,Default,,0000,0000,0000,,Another example is 3X minus\Nfive is less than or
Dialogue: 0,0:05:43.77,0:05:47.16,Default,,0000,0000,0000,,equal to 3 minus X.
Dialogue: 0,0:05:48.86,0:05:54.23,Default,,0000,0000,0000,,This time I need to add 5 to\Nboth sides which gives me 3X is
Dialogue: 0,0:05:54.23,0:05:56.02,Default,,0000,0000,0000,,less than or equal to.
Dialogue: 0,0:05:56.53,0:05:58.80,Default,,0000,0000,0000,,8 minus X.
Dialogue: 0,0:05:59.44,0:06:04.00,Default,,0000,0000,0000,,And then I need to add extra\Nboth sides, which gives me 4X
Dialogue: 0,0:06:04.00,0:06:06.11,Default,,0000,0000,0000,,less than or equal to 8.
Dialogue: 0,0:06:06.89,0:06:11.94,Default,,0000,0000,0000,,Finally, I can divide both sides\Nby two, which gives me X is less
Dialogue: 0,0:06:11.94,0:06:13.75,Default,,0000,0000,0000,,than or equal to two.
Dialogue: 0,0:06:14.98,0:06:16.16,Default,,0000,0000,0000,,And on the number line.
Dialogue: 0,0:06:18.83,0:06:22.54,Default,,0000,0000,0000,,X is less than or equal to two,\Nso we go this way.
Dialogue: 0,0:06:23.29,0:06:26.19,Default,,0000,0000,0000,,And this time I'm going to do a
Dialogue: 0,0:06:26.19,0:06:30.36,Default,,0000,0000,0000,,closed circle. This denotes that\NX can be equal to two.
Dialogue: 0,0:06:33.13,0:06:40.29,Default,,0000,0000,0000,,Now I'd like to look at\Nthe inequality minus 2X is
Dialogue: 0,0:06:40.29,0:06:42.24,Default,,0000,0000,0000,,greater than 4.
Dialogue: 0,0:06:43.26,0:06:46.64,Default,,0000,0000,0000,,In order to solve this\Ninequality, we're going to have
Dialogue: 0,0:06:46.64,0:06:49.01,Default,,0000,0000,0000,,to divide both sides by minus 2.
Dialogue: 0,0:06:51.93,0:06:56.74,Default,,0000,0000,0000,,So we get minus two X divided by\Nminus two is X.
Dialogue: 0,0:06:58.06,0:07:02.09,Default,,0000,0000,0000,,I've got to remember because I'm\Ndividing by a negative number to
Dialogue: 0,0:07:02.09,0:07:03.10,Default,,0000,0000,0000,,reverse the inequality.
Dialogue: 0,0:07:04.14,0:07:09.01,Default,,0000,0000,0000,,And four divided by minus\Ntwo is minus 2, so I get
Dialogue: 0,0:07:09.01,0:07:11.45,Default,,0000,0000,0000,,X is less than minus 2.
Dialogue: 0,0:07:14.39,0:07:17.33,Default,,0000,0000,0000,,There's often more than one way\Nto solve an inequality.
Dialogue: 0,0:07:18.55,0:07:21.54,Default,,0000,0000,0000,,And I can just solve this\None again by using a
Dialogue: 0,0:07:21.54,0:07:24.53,Default,,0000,0000,0000,,different method, so we have\N-2 X is greater than 4.
Dialogue: 0,0:07:25.89,0:07:28.79,Default,,0000,0000,0000,,If we add 2X to both sides we
Dialogue: 0,0:07:28.79,0:07:34.85,Default,,0000,0000,0000,,get. Zero is greater than\N4 + 2 X.
Dialogue: 0,0:07:36.70,0:07:42.59,Default,,0000,0000,0000,,And then if we subtract 4 from\Nboth sides, we get minus four is
Dialogue: 0,0:07:42.59,0:07:44.28,Default,,0000,0000,0000,,greater than two X.
Dialogue: 0,0:07:44.90,0:07:50.50,Default,,0000,0000,0000,,And we can divide through by two\Nagain getting minus two is
Dialogue: 0,0:07:50.50,0:07:51.90,Default,,0000,0000,0000,,greater than X.
Dialogue: 0,0:07:52.45,0:07:57.36,Default,,0000,0000,0000,,And saying that X is less than\Nminus two is the same thing as
Dialogue: 0,0:07:57.36,0:08:01.22,Default,,0000,0000,0000,,saying minus two is greater than\NX, so we've solved this
Dialogue: 0,0:08:01.22,0:08:04.38,Default,,0000,0000,0000,,inequality by do different\Nmethods. The second one avoids
Dialogue: 0,0:08:04.38,0:08:06.14,Default,,0000,0000,0000,,dividing by a negative number.
Dialogue: 0,0:08:07.76,0:08:13.91,Default,,0000,0000,0000,,In Equalities often appear in\Nconjunction with the modulus
Dialogue: 0,0:08:13.91,0:08:17.15,Default,,0000,0000,0000,,symbol. For instance.
Dialogue: 0,0:08:18.84,0:08:22.61,Default,,0000,0000,0000,,We say MoD X is less than two.
Dialogue: 0,0:08:23.70,0:08:27.41,Default,,0000,0000,0000,,The modular symbol denotes that\Nwe have to take the absolute
Dialogue: 0,0:08:27.41,0:08:31.79,Default,,0000,0000,0000,,value of X regardless of sign.\NThis is just the magnitude of X.
Dialogue: 0,0:08:33.47,0:08:36.56,Default,,0000,0000,0000,,And it is always\Npositive. So for
Dialogue: 0,0:08:36.56,0:08:39.66,Default,,0000,0000,0000,,instance, MoD 2 is\Nequal to 2.
Dialogue: 0,0:08:41.01,0:08:45.39,Default,,0000,0000,0000,,And MoD minus two is\Nalso equal to two.
Dialogue: 0,0:08:46.85,0:08:53.04,Default,,0000,0000,0000,,If the absolute value of X is\Nless than two, then X must lie
Dialogue: 0,0:08:53.04,0:08:58.78,Default,,0000,0000,0000,,between 2:00 and minus two. We\Nwrite minus two is less than X,
Dialogue: 0,0:08:58.78,0:09:00.55,Default,,0000,0000,0000,,is less than two.
Dialogue: 0,0:09:01.26,0:09:05.10,Default,,0000,0000,0000,,We can show this on the\Nnumber line.
Dialogue: 0,0:09:06.98,0:09:14.79,Default,,0000,0000,0000,,X has to lie between minus two\Nand two, but it can't be too
Dialogue: 0,0:09:14.79,0:09:22.37,Default,,0000,0000,0000,,itself. This shows the range\Nof values that ex can take.
Dialogue: 0,0:09:25.32,0:09:31.12,Default,,0000,0000,0000,,If MoD X is greater than or\Nequal to five, we have the
Dialogue: 0,0:09:31.12,0:09:36.47,Default,,0000,0000,0000,,absolute value of X must be\Ngreater than or equal to five,
Dialogue: 0,0:09:36.47,0:09:42.27,Default,,0000,0000,0000,,which means that X is going to\Nitself is going to be greater
Dialogue: 0,0:09:42.27,0:09:48.51,Default,,0000,0000,0000,,than or equal to five or less\Nthan or equal to minus five. We
Dialogue: 0,0:09:48.51,0:09:54.76,Default,,0000,0000,0000,,write X less than or equal to\Nminus five or X greater than or
Dialogue: 0,0:09:54.76,0:09:56.09,Default,,0000,0000,0000,,equal to 5.
Dialogue: 0,0:09:56.27,0:09:57.71,Default,,0000,0000,0000,,And on the number line.
Dialogue: 0,0:09:59.30,0:10:03.58,Default,,0000,0000,0000,,X can take the value 5, so we do\Na closed circle.
Dialogue: 0,0:10:04.90,0:10:08.00,Default,,0000,0000,0000,,And it can take the\Nvalue minus 5.
Dialogue: 0,0:10:10.21,0:10:15.87,Default,,0000,0000,0000,,Now I want to look at\Nanother slightly more
Dialogue: 0,0:10:15.87,0:10:17.76,Default,,0000,0000,0000,,complicated modulus one.
Dialogue: 0,0:10:18.89,0:10:21.62,Default,,0000,0000,0000,,We have MoD X minus 4.
Dialogue: 0,0:10:22.83,0:10:24.50,Default,,0000,0000,0000,,Less than three.
Dialogue: 0,0:10:25.39,0:10:30.33,Default,,0000,0000,0000,,The modulus sign shows that\Nthe absolute value of X minus
Dialogue: 0,0:10:30.33,0:10:35.72,Default,,0000,0000,0000,,four is less than three. This\Nmeans that X minus four must
Dialogue: 0,0:10:35.72,0:10:40.21,Default,,0000,0000,0000,,lie between minus three and\Nthree, so we write minus
Dialogue: 0,0:10:40.21,0:10:44.25,Default,,0000,0000,0000,,three less than X minus four\Nless than three.
Dialogue: 0,0:10:45.91,0:10:50.91,Default,,0000,0000,0000,,This is what we call a double\Ninequality of women's treated as
Dialogue: 0,0:10:50.91,0:10:55.92,Default,,0000,0000,0000,,two separate inequalities. So on\Nthe left we have minus three is
Dialogue: 0,0:10:55.92,0:10:58.00,Default,,0000,0000,0000,,less than X minus 4.
Dialogue: 0,0:11:00.22,0:11:06.96,Default,,0000,0000,0000,,By adding four to both sides, we\Nget one is less than X. On the
Dialogue: 0,0:11:06.96,0:11:11.44,Default,,0000,0000,0000,,right we have X minus four is\Nless than three.
Dialogue: 0,0:11:12.11,0:11:17.09,Default,,0000,0000,0000,,And again we had four to both\Nsides to get. X is less than 7.
Dialogue: 0,0:11:17.75,0:11:21.61,Default,,0000,0000,0000,,So the solution to this\Nparticular inequality is X is
Dialogue: 0,0:11:21.61,0:11:26.24,Default,,0000,0000,0000,,greater than One X is less\Nthan Seven. We write 1 less
Dialogue: 0,0:11:26.24,0:11:30.87,Default,,0000,0000,0000,,than X less than Seven, and\Nagain I'll show you that on
Dialogue: 0,0:11:30.87,0:11:32.03,Default,,0000,0000,0000,,the number line.
Dialogue: 0,0:11:34.51,0:11:38.48,Default,,0000,0000,0000,,X lies between one and Seven,\Nbut it can't be either.
Dialogue: 0,0:11:42.95,0:11:49.23,Default,,0000,0000,0000,,Now let's solve\NMoD. 5X. Minus 8
Dialogue: 0,0:11:49.23,0:11:55.51,Default,,0000,0000,0000,,is less than or\Nequal to 12.
Dialogue: 0,0:11:58.00,0:12:02.14,Default,,0000,0000,0000,,We're saying here that the\Nabsolute value of 5X minus 8 is
Dialogue: 0,0:12:02.14,0:12:04.21,Default,,0000,0000,0000,,less than or equal to 12.
Dialogue: 0,0:12:05.08,0:12:07.27,Default,,0000,0000,0000,,So 5X minus 8.
Dialogue: 0,0:12:07.82,0:12:09.46,Default,,0000,0000,0000,,Must be less than 12.
Dialogue: 0,0:12:10.85,0:12:13.02,Default,,0000,0000,0000,,Or greater than minus 12.
Dialogue: 0,0:12:13.81,0:12:20.61,Default,,0000,0000,0000,,We write minus 12 is less than\Nor equal to 5X minus 8.
Dialogue: 0,0:12:21.26,0:12:23.71,Default,,0000,0000,0000,,Is less than or equal to 12?
Dialogue: 0,0:12:25.03,0:12:30.20,Default,,0000,0000,0000,,Again, we have a double\Ninequality on the left, we have
Dialogue: 0,0:12:30.20,0:12:35.37,Default,,0000,0000,0000,,minus 12 is less than or equal\Nto 5X minus 8.
Dialogue: 0,0:12:36.48,0:12:42.18,Default,,0000,0000,0000,,We add it to both sides, which\Ngives us minus four is less than
Dialogue: 0,0:12:42.18,0:12:43.81,Default,,0000,0000,0000,,or equal to 5X.
Dialogue: 0,0:12:44.96,0:12:48.97,Default,,0000,0000,0000,,And then we divide both\Nsides by 5, which gives
Dialogue: 0,0:12:48.97,0:12:53.38,Default,,0000,0000,0000,,us minus four fifths is\Nless than or equal to X.
Dialogue: 0,0:12:54.46,0:12:58.71,Default,,0000,0000,0000,,On the right we have the\Ninequality 5X minus 8 is less
Dialogue: 0,0:12:58.71,0:13:00.48,Default,,0000,0000,0000,,than or equal to 12.
Dialogue: 0,0:13:01.48,0:13:06.63,Default,,0000,0000,0000,,So we write 5X minus 8 less than\Nor equal to 12.
Dialogue: 0,0:13:07.36,0:13:12.26,Default,,0000,0000,0000,,We had eight to both sides,\Nwhich gives us 5X is less than
Dialogue: 0,0:13:12.26,0:13:13.77,Default,,0000,0000,0000,,or equal to 20.
Dialogue: 0,0:13:14.51,0:13:18.37,Default,,0000,0000,0000,,And we divide both sides\Nby 5, which gives us X is
Dialogue: 0,0:13:18.37,0:13:20.31,Default,,0000,0000,0000,,less than or equal to 4.
Dialogue: 0,0:13:22.07,0:13:28.68,Default,,0000,0000,0000,,So our final answer is minus 4\Nover 5 is less than or equal to
Dialogue: 0,0:13:28.68,0:13:32.24,Default,,0000,0000,0000,,X. Which in turn is less\Nthan or equal to 4.
Dialogue: 0,0:13:33.44,0:13:35.83,Default,,0000,0000,0000,,And we can show this\Non the number line.
Dialogue: 0,0:13:37.19,0:13:40.01,Default,,0000,0000,0000,,Minus four fifths is about here.
Dialogue: 0,0:13:40.93,0:13:42.46,Default,,0000,0000,0000,,Let me go through to four.
Dialogue: 0,0:13:43.16,0:13:45.18,Default,,0000,0000,0000,,And because it's less than or
Dialogue: 0,0:13:45.18,0:13:48.86,Default,,0000,0000,0000,,equal to. We use\Na closed circle.
Dialogue: 0,0:13:50.70,0:13:54.68,Default,,0000,0000,0000,,In Equalities can be solved\Nvery easily using graphs,
Dialogue: 0,0:13:54.68,0:13:59.54,Default,,0000,0000,0000,,and if you're in any way\Nunsure about the algebra it
Dialogue: 0,0:13:59.54,0:14:05.73,Default,,0000,0000,0000,,can could be a good idea to\Ndo a graph to check. Let me
Dialogue: 0,0:14:05.73,0:14:07.94,Default,,0000,0000,0000,,show you how this works.
Dialogue: 0,0:14:09.70,0:14:15.36,Default,,0000,0000,0000,,We take the inequality 2X, plus\Nthree is less than 0.
Dialogue: 0,0:14:16.04,0:14:18.99,Default,,0000,0000,0000,,Now this inequality can be\Nsolved very easily doing
Dialogue: 0,0:14:18.99,0:14:20.96,Default,,0000,0000,0000,,algebra, but it makes a good
Dialogue: 0,0:14:20.96,0:14:27.31,Default,,0000,0000,0000,,example. The first thing that we\Nneed to do is to draw the graph
Dialogue: 0,0:14:27.31,0:14:29.72,Default,,0000,0000,0000,,of Y equals 2X plus 3.
Dialogue: 0,0:14:32.18,0:14:33.64,Default,,0000,0000,0000,,And I've got this graph here.
Dialogue: 0,0:14:34.20,0:14:39.74,Default,,0000,0000,0000,,Note that it's the equation of\Na straight line.
Dialogue: 0,0:14:40.44,0:14:43.82,Default,,0000,0000,0000,,It has a slope of two\Nand then intercept on
Dialogue: 0,0:14:43.82,0:14:45.51,Default,,0000,0000,0000,,the Y axis of three.
Dialogue: 0,0:14:47.45,0:14:51.28,Default,,0000,0000,0000,,On the X axis.
Dialogue: 0,0:14:52.46,0:14:56.31,Default,,0000,0000,0000,,Why is equal to 0 so that\Nwhere the line cuts the X
Dialogue: 0,0:14:56.31,0:14:58.08,Default,,0000,0000,0000,,axis Y is equal to 0?
Dialogue: 0,0:14:59.28,0:15:01.63,Default,,0000,0000,0000,,Above the X axis Y is greater
Dialogue: 0,0:15:01.63,0:15:06.39,Default,,0000,0000,0000,,than 0. And below the X axis Y\Nis less than 0.
Dialogue: 0,0:15:08.26,0:15:11.98,Default,,0000,0000,0000,,So when we say that we want 2X\Nplus three less than 0.
Dialogue: 0,0:15:13.42,0:15:17.20,Default,,0000,0000,0000,,On this graph, that means why is\Nless than zero, so we're looking
Dialogue: 0,0:15:17.20,0:15:20.40,Default,,0000,0000,0000,,for the points where the line is\Nbelow the X axis.
Dialogue: 0,0:15:21.09,0:15:25.68,Default,,0000,0000,0000,,In other words, where X is less\Nthan minus one and a half, and
Dialogue: 0,0:15:25.68,0:15:27.65,Default,,0000,0000,0000,,this is the solution to the
Dialogue: 0,0:15:27.65,0:15:35.24,Default,,0000,0000,0000,,inequality. And we can mark\Nthis on the graph using the
Dialogue: 0,0:15:35.24,0:15:39.13,Default,,0000,0000,0000,,X axis as the number line.
Dialogue: 0,0:15:39.85,0:15:46.33,Default,,0000,0000,0000,,This technique can also be\Nused with modulus inequalities
Dialogue: 0,0:15:46.33,0:15:52.81,Default,,0000,0000,0000,,and here using a graph\Ncan be very helpful.
Dialogue: 0,0:15:53.75,0:15:56.44,Default,,0000,0000,0000,,Take for example the inequality.
Dialogue: 0,0:15:57.01,0:16:00.69,Default,,0000,0000,0000,,MoD X minus two is less than 0.
Dialogue: 0,0:16:01.82,0:16:08.15,Default,,0000,0000,0000,,Again, we need to plot the graph\Nof Y equals MoD X minus 2.
Dialogue: 0,0:16:08.72,0:16:14.92,Default,,0000,0000,0000,,This is the graph of Y equals\NMoD X minus 2.
Dialogue: 0,0:16:15.75,0:16:18.24,Default,,0000,0000,0000,,For those of you who are not\Nfamiliar with modulus functions,
Dialogue: 0,0:16:18.24,0:16:19.59,Default,,0000,0000,0000,,it might look a little bit
Dialogue: 0,0:16:19.59,0:16:24.44,Default,,0000,0000,0000,,strange. On the right we have\Npart of the graph of Y equals X
Dialogue: 0,0:16:24.44,0:16:29.60,Default,,0000,0000,0000,,minus 2. And on the left,\Nwhere X is less than zero, we
Dialogue: 0,0:16:29.60,0:16:33.71,Default,,0000,0000,0000,,have part of the graph of Y\Nequals minus X minus two.
Dialogue: 0,0:16:33.71,0:16:37.13,Default,,0000,0000,0000,,This is because the modulus\Nfunction changes the sign of
Dialogue: 0,0:16:37.13,0:16:38.84,Default,,0000,0000,0000,,X when X is negative.
Dialogue: 0,0:16:40.66,0:16:45.58,Default,,0000,0000,0000,,Again, we're looking for MoD X.\NMinus two is less than 0.
Dialogue: 0,0:16:46.76,0:16:52.12,Default,,0000,0000,0000,,So we want the places where Y is\Nless than zero, which is between
Dialogue: 0,0:16:52.12,0:16:57.10,Default,,0000,0000,0000,,X equals minus two and X equals\N+2, and again this is the
Dialogue: 0,0:16:57.10,0:16:58.63,Default,,0000,0000,0000,,solution to our problem.
Dialogue: 0,0:16:59.46,0:17:05.21,Default,,0000,0000,0000,,So we say minus two less than\NX less than two.
Dialogue: 0,0:17:05.92,0:17:10.53,Default,,0000,0000,0000,,Again, we can mark this on the\Ngraph using the X axis as the
Dialogue: 0,0:17:10.53,0:17:15.29,Default,,0000,0000,0000,,number line. Quadratic\Ninequalities need
Dialogue: 0,0:17:15.29,0:17:22.13,Default,,0000,0000,0000,,handling with care.\NLet's solve X
Dialogue: 0,0:17:22.13,0:17:28.97,Default,,0000,0000,0000,,squared minus three\NX +2 is
Dialogue: 0,0:17:28.97,0:17:32.39,Default,,0000,0000,0000,,greater than 0.
Dialogue: 0,0:17:35.61,0:17:38.73,Default,,0000,0000,0000,,Note that all the terms are on\Nthe left hand side.
Dialogue: 0,0:17:39.24,0:17:42.87,Default,,0000,0000,0000,,And on the right hand side we\Njust had zero, exactly as with
Dialogue: 0,0:17:42.87,0:17:43.98,Default,,0000,0000,0000,,the quadratic equation before
Dialogue: 0,0:17:43.98,0:17:47.65,Default,,0000,0000,0000,,you solve it. This expression
Dialogue: 0,0:17:47.65,0:17:53.75,Default,,0000,0000,0000,,factorizes too. X minus\Ntwo X minus one.
Dialogue: 0,0:17:54.53,0:17:58.31,Default,,0000,0000,0000,,Now this is a quadratic\Nequation. We would simply say
Dialogue: 0,0:17:58.31,0:18:02.47,Default,,0000,0000,0000,,right X equals 2 or X equals 1\Nand that's it.
Dialogue: 0,0:18:03.25,0:18:04.68,Default,,0000,0000,0000,,But we've got a bit more work to
Dialogue: 0,0:18:04.68,0:18:10.12,Default,,0000,0000,0000,,do here. Weather this expression\Nis greater than zero is going to
Dialogue: 0,0:18:10.12,0:18:15.45,Default,,0000,0000,0000,,depend on the sign of each of\Nthese two factors. We sort this
Dialogue: 0,0:18:15.45,0:18:17.50,Default,,0000,0000,0000,,out by using a grid.
Dialogue: 0,0:18:18.24,0:18:24.74,Default,,0000,0000,0000,,The points\Nthat were
Dialogue: 0,0:18:24.74,0:18:31.37,Default,,0000,0000,0000,,checks equals.\NX minus 2 equals 0 and X minus
Dialogue: 0,0:18:31.37,0:18:35.39,Default,,0000,0000,0000,,one equals 0 and marked in, so\Nthis is one and two.
Dialogue: 0,0:18:36.17,0:18:39.58,Default,,0000,0000,0000,,We put the two factors on the
Dialogue: 0,0:18:39.58,0:18:42.70,Default,,0000,0000,0000,,left. And their product.
Dialogue: 0,0:18:43.28,0:18:47.00,Default,,0000,0000,0000,,Now.
Dialogue: 0,0:18:48.21,0:18:53.70,Default,,0000,0000,0000,,When X is less than one, both X\Nminus one and X minus two are
Dialogue: 0,0:18:53.70,0:18:55.16,Default,,0000,0000,0000,,going to be negative.
Dialogue: 0,0:18:56.58,0:18:59.95,Default,,0000,0000,0000,,So when you multiply them\Ntogether, their product is going
Dialogue: 0,0:18:59.95,0:19:00.96,Default,,0000,0000,0000,,to be positive.
Dialogue: 0,0:19:03.39,0:19:05.52,Default,,0000,0000,0000,,When X is greater than one but
Dialogue: 0,0:19:05.52,0:19:09.69,Default,,0000,0000,0000,,less than two. X minus one is\Ngoing to be positive.
Dialogue: 0,0:19:10.60,0:19:13.10,Default,,0000,0000,0000,,But X minus two is going to be
Dialogue: 0,0:19:13.10,0:19:15.35,Default,,0000,0000,0000,,negative. So when you multiply
Dialogue: 0,0:19:15.35,0:19:17.39,Default,,0000,0000,0000,,them together. The product will
Dialogue: 0,0:19:17.39,0:19:23.42,Default,,0000,0000,0000,,be negative. Finally, when X is\Ngreater than two, both X minus
Dialogue: 0,0:19:23.42,0:19:26.56,Default,,0000,0000,0000,,one and X minus two will be
Dialogue: 0,0:19:26.56,0:19:30.28,Default,,0000,0000,0000,,positive. And if you multiply\Nthem together, their product
Dialogue: 0,0:19:30.28,0:19:31.58,Default,,0000,0000,0000,,will also be positive.
Dialogue: 0,0:19:34.07,0:19:35.80,Default,,0000,0000,0000,,We are looking for.
Dialogue: 0,0:19:36.30,0:19:39.90,Default,,0000,0000,0000,,X minus two times X minus one to\Nbe greater than 0.
Dialogue: 0,0:19:40.89,0:19:42.62,Default,,0000,0000,0000,,This occurs when it's positive.
Dialogue: 0,0:19:43.50,0:19:47.14,Default,,0000,0000,0000,,And our grid shows that this\Nhappens when X is less than one.
Dialogue: 0,0:19:47.64,0:19:49.87,Default,,0000,0000,0000,,Or when X is greater than two?
Dialogue: 0,0:19:50.45,0:19:52.42,Default,,0000,0000,0000,,So we write in our answer.
Dialogue: 0,0:19:53.66,0:20:00.85,Default,,0000,0000,0000,,Which is X is less than one\Nor X is greater than two.
Dialogue: 0,0:20:03.95,0:20:06.59,Default,,0000,0000,0000,,And on the number line.
Dialogue: 0,0:20:07.21,0:20:09.39,Default,,0000,0000,0000,,X must be less than one.
Dialogue: 0,0:20:09.98,0:20:12.54,Default,,0000,0000,0000,,So I put a circle to show\Nthat it can't be 1.
Dialogue: 0,0:20:14.28,0:20:16.52,Default,,0000,0000,0000,,And X can also be greater\Nthan two.
Dialogue: 0,0:20:20.05,0:20:23.98,Default,,0000,0000,0000,,Here's another
Dialogue: 0,0:20:23.98,0:20:30.12,Default,,0000,0000,0000,,quadratic. Minus two\NX squared plus 5X
Dialogue: 0,0:20:30.12,0:20:35.48,Default,,0000,0000,0000,,plus 12 is greater\Nthan or equal to 0.
Dialogue: 0,0:20:36.57,0:20:40.67,Default,,0000,0000,0000,,I don't like having a negative\Ncoefficient of X squared, so I'm
Dialogue: 0,0:20:40.67,0:20:44.09,Default,,0000,0000,0000,,going to multiply this whole\Nthing through by minus one,
Dialogue: 0,0:20:44.09,0:20:47.51,Default,,0000,0000,0000,,remembering to change the\Ndirection of the inequality as I
Dialogue: 0,0:20:47.51,0:20:48.88,Default,,0000,0000,0000,,do. This gives us.
Dialogue: 0,0:20:49.41,0:20:57.28,Default,,0000,0000,0000,,Two X squared minus 5X minus 12\Nis less than or equal to 0.
Dialogue: 0,0:20:58.68,0:21:04.91,Default,,0000,0000,0000,,This expression factorizes to 2X\Nplus three times X minus four,
Dialogue: 0,0:21:04.91,0:21:08.87,Default,,0000,0000,0000,,so that is less than or equal
Dialogue: 0,0:21:08.87,0:21:12.96,Default,,0000,0000,0000,,to 0. Again, I'm going to\Ndo a grid.
Dialogue: 0,0:21:18.15,0:21:25.59,Default,,0000,0000,0000,,This factor is zero\Nwhen X is minus
Dialogue: 0,0:21:25.59,0:21:28.38,Default,,0000,0000,0000,,three over 2.
Dialogue: 0,0:21:29.45,0:21:31.86,Default,,0000,0000,0000,,This fact is zero when X is 4.
Dialogue: 0,0:21:32.77,0:21:35.70,Default,,0000,0000,0000,,We write in the two factors.
Dialogue: 0,0:21:36.38,0:21:39.94,Default,,0000,0000,0000,,And we right in the product.
Dialogue: 0,0:21:43.46,0:21:50.53,Default,,0000,0000,0000,,When X is less than minus three\Nover 2, both 2X plus three and
Dialogue: 0,0:21:50.53,0:21:53.06,Default,,0000,0000,0000,,X minus four and negative.
Dialogue: 0,0:21:53.86,0:21:56.11,Default,,0000,0000,0000,,So their product is positive.
Dialogue: 0,0:21:57.58,0:22:01.35,Default,,0000,0000,0000,,When X lies between minus three\Nover two and four.
Dialogue: 0,0:22:02.54,0:22:04.67,Default,,0000,0000,0000,,2X plus three is positive.
Dialogue: 0,0:22:05.41,0:22:09.59,Default,,0000,0000,0000,,But X minus four is still\Nnegative, so their product
Dialogue: 0,0:22:09.59,0:22:10.43,Default,,0000,0000,0000,,is negative.
Dialogue: 0,0:22:11.48,0:22:16.80,Default,,0000,0000,0000,,When X is greater than four,\Nboth 2X plus three and X minus
Dialogue: 0,0:22:16.80,0:22:18.02,Default,,0000,0000,0000,,four are positive.
Dialogue: 0,0:22:18.59,0:22:20.18,Default,,0000,0000,0000,,So their product is positive.
Dialogue: 0,0:22:20.78,0:22:26.58,Default,,0000,0000,0000,,We are looking for 2X plus three\Ntimes X minus four to be less
Dialogue: 0,0:22:26.58,0:22:28.65,Default,,0000,0000,0000,,than or equal to 0.
Dialogue: 0,0:22:29.33,0:22:33.11,Default,,0000,0000,0000,,In other words, this expression\Nhas to be either 0 or negative.
Dialogue: 0,0:22:34.30,0:22:35.27,Default,,0000,0000,0000,,This occurs.
Dialogue: 0,0:22:36.52,0:22:41.82,Default,,0000,0000,0000,,When X lies between minus three\Nover two and four, and it can
Dialogue: 0,0:22:41.82,0:22:47.13,Default,,0000,0000,0000,,equal either number. So we have\Nminus three over 2 is less than
Dialogue: 0,0:22:47.13,0:22:51.62,Default,,0000,0000,0000,,or equal to X is less than or\Nequal to 4.
Dialogue: 0,0:22:53.89,0:22:56.37,Default,,0000,0000,0000,,And on the number line.
Dialogue: 0,0:22:58.22,0:23:00.33,Default,,0000,0000,0000,,Minus three over 2 is here.
Dialogue: 0,0:23:01.76,0:23:05.74,Default,,0000,0000,0000,,Four is here.
Dialogue: 0,0:23:08.84,0:23:12.04,Default,,0000,0000,0000,,And I've done filled\Ncircles because we have
Dialogue: 0,0:23:12.04,0:23:14.04,Default,,0000,0000,0000,,less than or equal to.
Dialogue: 0,0:23:17.26,0:23:22.78,Default,,0000,0000,0000,,Quadratic inequalities can\Nalso be solved graphically.
Dialogue: 0,0:23:22.78,0:23:30.67,Default,,0000,0000,0000,,Let's solve X squared minus\Nthree X +2 is greater
Dialogue: 0,0:23:30.67,0:23:32.25,Default,,0000,0000,0000,,than 0.
Dialogue: 0,0:23:34.13,0:23:38.71,Default,,0000,0000,0000,,As with the linear equalities\Ninequalities, we have to plot
Dialogue: 0,0:23:38.71,0:23:43.75,Default,,0000,0000,0000,,the graph of Y equals X squared\Nminus three X +2.
Dialogue: 0,0:23:44.65,0:23:51.53,Default,,0000,0000,0000,,This factorizes to give Y equals\NX minus one times X minus 2.
Dialogue: 0,0:23:52.80,0:23:54.60,Default,,0000,0000,0000,,The graph looks like this.
Dialogue: 0,0:23:55.96,0:24:01.17,Default,,0000,0000,0000,,Because it's a quadratic, it's a\Nparabola. Are U shaped curve?
Dialogue: 0,0:24:02.21,0:24:04.28,Default,,0000,0000,0000,,And it crosses the X axis where
Dialogue: 0,0:24:04.28,0:24:08.73,Default,,0000,0000,0000,,X equals 1. Because of the\Nfactor X minus one and where
Dialogue: 0,0:24:08.73,0:24:12.14,Default,,0000,0000,0000,,X equals 2 because of the\Nfactor X minus 2.
Dialogue: 0,0:24:13.49,0:24:18.96,Default,,0000,0000,0000,,Now we're looking for X squared\Nminus three X +2 to be greater
Dialogue: 0,0:24:18.96,0:24:23.66,Default,,0000,0000,0000,,than 0. This is where Y\Nis greater than zero. In
Dialogue: 0,0:24:23.66,0:24:27.04,Default,,0000,0000,0000,,other words, the part of\Nthe graph that is above
Dialogue: 0,0:24:27.04,0:24:31.10,Default,,0000,0000,0000,,the X axis, which are the\Ntwo arms of the you here.
Dialogue: 0,0:24:32.71,0:24:36.02,Default,,0000,0000,0000,,This occurs where X is less than
Dialogue: 0,0:24:36.02,0:24:41.22,Default,,0000,0000,0000,,one. And where X is greater\Nthan two, so we can write
Dialogue: 0,0:24:41.22,0:24:43.06,Default,,0000,0000,0000,,that in as our solution.
Dialogue: 0,0:24:46.14,0:24:52.04,Default,,0000,0000,0000,,And we can mark this\Nin using the X axis
Dialogue: 0,0:24:52.04,0:24:54.40,Default,,0000,0000,0000,,as the number line.
Dialogue: 0,0:24:55.60,0:25:00.22,Default,,0000,0000,0000,,I'll\Ndo
Dialogue: 0,0:25:00.22,0:25:04.84,Default,,0000,0000,0000,,one\Nmore
Dialogue: 0,0:25:04.84,0:25:07.16,Default,,0000,0000,0000,,quadratic
Dialogue: 0,0:25:07.16,0:25:09.47,Default,,0000,0000,0000,,inequality.
Dialogue: 0,0:25:10.47,0:25:14.04,Default,,0000,0000,0000,,X squared Minus X
Dialogue: 0,0:25:14.04,0:25:18.42,Default,,0000,0000,0000,,minus 6. So less than or\Nequal to 0.
Dialogue: 0,0:25:22.68,0:25:27.15,Default,,0000,0000,0000,,Again, we need to plot\Nthe graph of Y equals X
Dialogue: 0,0:25:27.15,0:25:29.18,Default,,0000,0000,0000,,squared minus X minus 6.
Dialogue: 0,0:25:30.36,0:25:32.06,Default,,0000,0000,0000,,The expression factorizes.
Dialogue: 0,0:25:32.83,0:25:35.100,Default,,0000,0000,0000,,To X minus three.
Dialogue: 0,0:25:36.07,0:25:40.03,Default,,0000,0000,0000,,X +2 And the graph
Dialogue: 0,0:25:40.03,0:25:46.76,Default,,0000,0000,0000,,looks like this. Similar\Nto the previous
Dialogue: 0,0:25:46.76,0:25:48.04,Default,,0000,0000,0000,,graph.
Dialogue: 0,0:25:49.21,0:25:54.72,Default,,0000,0000,0000,,We have The factor X +2 the line\Ncrosses the point at X equals
Dialogue: 0,0:25:54.72,0:25:58.83,Default,,0000,0000,0000,,minus two and for the factor X\Nminus three, the curve crosses
Dialogue: 0,0:25:58.83,0:26:00.89,Default,,0000,0000,0000,,the point at X equals 3.
Dialogue: 0,0:26:01.75,0:26:06.05,Default,,0000,0000,0000,,And we're looking for where X\Nsquared minus X minus six is
Dialogue: 0,0:26:06.05,0:26:08.19,Default,,0000,0000,0000,,less than or equal to 0.
Dialogue: 0,0:26:09.47,0:26:14.19,Default,,0000,0000,0000,,In other words, why must lie on\Nthe X axis or below it?
Dialogue: 0,0:26:14.92,0:26:19.51,Default,,0000,0000,0000,,This part of the curve and that\Noccurs between the points of X
Dialogue: 0,0:26:19.51,0:26:24.80,Default,,0000,0000,0000,,equals minus two and X equals 3.\NSo we can say that minus two is
Dialogue: 0,0:26:24.80,0:26:29.75,Default,,0000,0000,0000,,less than or equal to X, which\Nis less than or equal to 3.
Dialogue: 0,0:26:31.26,0:26:36.75,Default,,0000,0000,0000,,And we can put this in again\Nusing the X axis is the
Dialogue: 0,0:26:36.75,0:26:40.97,Default,,0000,0000,0000,,number line from minus 2\Nusing a closed circle because
Dialogue: 0,0:26:40.97,0:26:43.92,Default,,0000,0000,0000,,2 - 2 is included to +3.