WEBVTT
00:00:00.000 --> 00:00:06.448 align:middle line:90%
00:00:06.448 --> 00:00:09.920 align:middle line:90%
[MUSIC PLAYING]
00:00:09.920 --> 00:00:12.910 align:middle line:90%
00:00:12.910 --> 00:00:15.230 align:middle line:90%
Hello, and welcome back.
00:00:15.230 --> 00:00:19.670 align:middle line:84%
We hope that you tried to
solve the exercises by yourself
00:00:19.670 --> 00:00:22.400 align:middle line:90%
because it's your turn now.
00:00:22.400 --> 00:00:25.070 align:middle line:84%
We're discussing inequalities
with exponentials
00:00:25.070 --> 00:00:26.030 align:middle line:90%
and logarithms.
00:00:26.030 --> 00:00:29.000 align:middle line:84%
And in the first
exercise, we have
00:00:29.000 --> 00:00:34.010 align:middle line:84%
to say if these inequalities
are true or false.
00:00:34.010 --> 00:00:37.220 align:middle line:84%
Let us begin with the first
one, and we express everything
00:00:37.220 --> 00:00:39.080 align:middle line:84%
in terms of the
natural logarithm.
00:00:39.080 --> 00:00:45.320 align:middle line:84%
So logarithm to base 2
of 3 is natural logarithm
00:00:45.320 --> 00:00:48.830 align:middle line:84%
of 3 divided by
natural logarithm of 2.
00:00:48.830 --> 00:00:53.270 align:middle line:84%
And the inequality is
equivalent to this quotient,
00:00:53.270 --> 00:00:56.960 align:middle line:84%
strictly less than the
natural logarithm of 2
00:00:56.960 --> 00:01:02.240 align:middle line:84%
divided by natural
logarithm of 3.
00:01:02.240 --> 00:01:03.960 align:middle line:90%
And this is equivalent--
00:01:03.960 --> 00:01:07.460 align:middle line:84%
notice that all the
members here are strictly
00:01:07.460 --> 00:01:10.910 align:middle line:84%
positive because they
are greater than 1.
00:01:10.910 --> 00:01:16.600 align:middle line:84%
So we multiply everything
by natural logarithm of 3
00:01:16.600 --> 00:01:18.460 align:middle line:84%
and natural logarithm
of 2, and we
00:01:18.460 --> 00:01:21.280 align:middle line:84%
get that this is
equivalent to the fact
00:01:21.280 --> 00:01:24.340 align:middle line:84%
that natural logarithm
of 3 to the square
00:01:24.340 --> 00:01:27.700 align:middle line:84%
is strictly less than the
natural logarithm of 2
00:01:27.700 --> 00:01:30.110 align:middle line:90%
to the square.
00:01:30.110 --> 00:01:35.380 align:middle line:84%
This is equivalent to the fact
that natural logarithm of 3
00:01:35.380 --> 00:01:39.700 align:middle line:84%
is strictly less than the
natural logarithm of 2.
00:01:39.700 --> 00:01:43.880 align:middle line:84%
But the natural logarithm
is strictly increasing,
00:01:43.880 --> 00:01:44.810 align:middle line:90%
so this is false.
00:01:44.810 --> 00:01:49.280 align:middle line:90%
00:01:49.280 --> 00:01:53.260 align:middle line:84%
So let us check the
second inequality.
00:01:53.260 --> 00:01:58.870 align:middle line:84%
The second inequality is
equivalent to the fact--
00:01:58.870 --> 00:02:06.790 align:middle line:84%
well, here, we can invert
and apply 2 to the power
00:02:06.790 --> 00:02:09.949 align:middle line:84%
to both members
of the inequality.
00:02:09.949 --> 00:02:14.710 align:middle line:84%
So think that logarithm 2 base
2 of 3 is strictly less than 3/2
00:02:14.710 --> 00:02:19.660 align:middle line:84%
is equivalent to the fact
that 3 is strictly less than 2
00:02:19.660 --> 00:02:22.960 align:middle line:90%
to the 3/2.
00:02:22.960 --> 00:02:29.460 align:middle line:84%
And this is equivalent,
we apply the power 2 to 3
00:02:29.460 --> 00:02:32.910 align:middle line:84%
to the square strictly
less than 2 to the cube.
00:02:32.910 --> 00:02:36.330 align:middle line:90%
00:02:36.330 --> 00:02:41.485 align:middle line:84%
So this is 9 and this is 8,
so this is, again, false.
00:02:41.485 --> 00:02:44.470 align:middle line:90%
00:02:44.470 --> 00:02:49.270 align:middle line:84%
Let us look at the
third inequality.
00:02:49.270 --> 00:02:54.970 align:middle line:84%
Again, here, we apply
3 to the power--
00:02:54.970 --> 00:02:57.040 align:middle line:90%
so 3 to something--
00:02:57.040 --> 00:03:00.280 align:middle line:84%
on both sides of the
inequality, and the inequality
00:03:00.280 --> 00:03:07.690 align:middle line:84%
is equivalent to 2 strictly
less than 3 to the power 2/3.
00:03:07.690 --> 00:03:11.110 align:middle line:84%
Then, we take the
power 3 to both sides
00:03:11.110 --> 00:03:15.080 align:middle line:84%
of the inequality, which is a
strictly increasing function.
00:03:15.080 --> 00:03:19.570 align:middle line:84%
So this is equivalent to 2 to
the cube strictly less than 3
00:03:19.570 --> 00:03:20.890 align:middle line:90%
to the square.
00:03:20.890 --> 00:03:24.210 align:middle line:90%
And this is right, yes.
00:03:24.210 --> 00:03:24.710 align:middle line:90%
True.
00:03:24.710 --> 00:03:28.566 align:middle line:90%
00:03:28.566 --> 00:03:33.910 align:middle line:90%
And this ends exercise 1.
00:03:33.910 --> 00:03:41.710 align:middle line:84%
In exercise 2, we want
to solve an inequality.
00:03:41.710 --> 00:03:48.810 align:middle line:84%
Well, let us find immediately
the domain of the inequality,
00:03:48.810 --> 00:03:54.300 align:middle line:84%
which is --well, we
cannot divide by 0.
00:03:54.300 --> 00:03:58.820 align:middle line:84%
And, well, 1/3 to
something is defined on R,
00:03:58.820 --> 00:04:06.590 align:middle line:84%
so the domain is the set
of real numbers without 0.
00:04:06.590 --> 00:04:09.920 align:middle line:90%
00:04:09.920 --> 00:04:18.180 align:middle line:84%
Now, the inequality
becomes 1/3 to minus 7
00:04:18.180 --> 00:04:24.340 align:middle line:84%
divided by x strictly
greater than 1/3 to x plus 8.
00:04:24.340 --> 00:04:30.330 align:middle line:84%
Now, let us recall
that 1/3 to something
00:04:30.330 --> 00:04:31.445 align:middle line:90%
is a decreasing function.
00:04:31.445 --> 00:04:38.380 align:middle line:90%
00:04:38.380 --> 00:04:42.340 align:middle line:84%
So this inequality is
equivalent to the fact
00:04:42.340 --> 00:04:46.120 align:middle line:84%
that the exponents satisfy the
opposite inequality-- that is,
00:04:46.120 --> 00:04:54.360 align:middle line:84%
minus 7 divided by x
strictly less than x plus 8.
00:04:54.360 --> 00:04:59.310 align:middle line:84%
And this is
equivalent to x plus 8
00:04:59.310 --> 00:05:03.610 align:middle line:84%
plus 7 divided by x
strictly greater than 0.
00:05:03.610 --> 00:05:06.540 align:middle line:84%
Now, we put everything
divided by x.
00:05:06.540 --> 00:05:13.260 align:middle line:84%
So this is equivalent to x
squared plus 8 x plus 7 divided
00:05:13.260 --> 00:05:15.570 align:middle line:90%
by x strictly greater than 0.
00:05:15.570 --> 00:05:19.800 align:middle line:84%
So it is enough
to understand what
00:05:19.800 --> 00:05:21.690 align:middle line:90%
is the sign of this quotient.
00:05:21.690 --> 00:05:24.780 align:middle line:84%
So let us look at the
zeros of the polynomial
00:05:24.780 --> 00:05:28.980 align:middle line:90%
x squared plus 8 x plus 7.
00:05:28.980 --> 00:05:32.700 align:middle line:90%
Well, it is a second-degree.
00:05:32.700 --> 00:05:39.330 align:middle line:84%
Zeros of x squared
plus 8 x plus 7.
00:05:39.330 --> 00:05:46.866 align:middle line:84%
Well, the reduced discriminant
is 4 to the square minus 7.
00:05:46.866 --> 00:05:52.160 align:middle line:84%
It is 16 minus 7,
9, 3 to the square.
00:05:52.160 --> 00:06:00.800 align:middle line:84%
And so the roots are x equals
to minus 4 plus or minus 3.
00:06:00.800 --> 00:06:11.430 align:middle line:84%
That is, we get equals to minus
4 minus 3, minus 7, or minus 1.
00:06:11.430 --> 00:06:13.550 align:middle line:90%
So we get two roots.
00:06:13.550 --> 00:06:18.080 align:middle line:84%
Now, let us prepare
the table of the signs
00:06:18.080 --> 00:06:21.275 align:middle line:84%
of the polynomial second
degree x squared plus 8
00:06:21.275 --> 00:06:22.970 align:middle line:90%
x plus 7 and x.
00:06:22.970 --> 00:06:27.630 align:middle line:84%
The important values are
minus 7, minus 1, and 0.
00:06:27.630 --> 00:06:32.020 align:middle line:90%
So let us write here the table.
00:06:32.020 --> 00:06:35.170 align:middle line:90%
So we write the values of x.
00:06:35.170 --> 00:06:42.590 align:middle line:84%
So we've got minus
7, minus 1, 0.
00:06:42.590 --> 00:06:51.850 align:middle line:84%
And here, we write the sign
of x squared plus 8 x plus 7
00:06:51.850 --> 00:06:53.320 align:middle line:90%
with this first line.
00:06:53.320 --> 00:06:57.050 align:middle line:84%
Whereas, in the second
line, the sign of x.
00:06:57.050 --> 00:06:58.830 align:middle line:84%
And here, the sign
of the quotient x
00:06:58.830 --> 00:07:03.930 align:middle line:84%
squared plus 8 x
plus 7 divided by x.
00:07:03.930 --> 00:07:09.460 align:middle line:84%
Now, the zeros are
minus 7 and minus 1,
00:07:09.460 --> 00:07:11.890 align:middle line:84%
So this is a
second-degree polynomial.
00:07:11.890 --> 00:07:20.000 align:middle line:84%
It's positive out of the roots
and negative inside the roots.
00:07:20.000 --> 00:07:25.230 align:middle line:84%
x, of course, is
0 in 0, positive
00:07:25.230 --> 00:07:27.390 align:middle line:84%
on the right-hand side,
and negative here.
00:07:27.390 --> 00:07:30.720 align:middle line:90%
00:07:30.720 --> 00:07:33.480 align:middle line:84%
The quotient is
not defined here.
00:07:33.480 --> 00:07:35.520 align:middle line:90%
It is 0 here and 0 here.
00:07:35.520 --> 00:07:38.160 align:middle line:84%
And then, we use
the rule of signs.
00:07:38.160 --> 00:07:43.420 align:middle line:90%
It is minus, plus, minus, plus.
00:07:43.420 --> 00:07:49.780 align:middle line:84%
So we want the quotient to
be strictly positive, so here
00:07:49.780 --> 00:07:50.930 align:middle line:90%
and here.
00:07:50.930 --> 00:07:52.840 align:middle line:90%
And x to belong to the domain--
00:07:52.840 --> 00:07:54.471 align:middle line:90%
that is, x different from 0.
00:07:54.471 --> 00:07:54.970 align:middle line:90%
OK.
00:07:54.970 --> 00:07:56.230 align:middle line:90%
Here we are.
00:07:56.230 --> 00:08:04.360 align:middle line:84%
So the solution is the interval
from minus 7 to minus 1
00:08:04.360 --> 00:08:09.350 align:middle line:84%
union with the interval
from 0 to plus infinity.
00:08:09.350 --> 00:08:14.040 align:middle line:84%
It is the solution
to exercise two.
00:08:14.040 --> 00:08:18.155 align:middle line:90%