Inequalities

This is a guide on inequalities.

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Inequalities

We recall that we write:
\(a > 0\)
if a is positive. If a,b are two real numbers, we shall write:
\(a > b\)
instead of
\(a - b > 0\)
We shall write
\(a < 0\)
instead of 
\(-a > 0\)
and also
\(b < a\)
instead of 
\(a > b\)

We have
\(3 > 2\)
because
\(3 - 2 = 1 > 0\)
We have
\(-1 > -2\)
because
\(-1 + 2 = 1 > 0\)

In the geometric representation of numbers on the line, the relation:
\(a > b\)
means that a lies to the right of b. We see that -1 lies to the right of -2 in
the example.

We shall write:
\(a \geq b\)
to mean a is greater than or equal to b.
\(3 \geq 2\)
and
\(3 \geq 3\)
are both true inequalitiews.

We wish to show that the inequality:
\(2x - 4 > 5\)
is equivalent to an inequality of type:
\(x > a\)
or
\(x < b\)
It is then equivalent to:
\(2x > 5 + 4 = 9\)
which is equivalent to
\(x > \frac{9}{2}\)

Suppose that x is a number such that:
\(\frac{3x + 5}{x - 4} < 2\)
We wish to find equivalent conditions under which this is true, expressed by
simpler inequalities like
\(x > a\)
or
\(x < b\)
Note that the quotient on the left makes no sense if
\(x = 4\)
It is then antural to consider the two cases separately:
\(x > 4\)
and 
\(x < 4\)
Suppose that 
\(x > 4\)
Then
\(x - 4 > 0\)
and our inequality is equivalent to
\(3x + 5 < 2(x - 4) = 2x - 8\)
This is equivalent to:
\(3x - 2x < -8 - 5\)
or
\(x < -13\)

Now assume that
\(x < 4\)
Then:
\(x - 4 < 0\)
and 
\(x - 4\)
is negative. We multiply both sides of our inequality by:
\(x - 4\)
and reverse the inequality. The inequality is equivalent to:
\(3x + 5 > 2x(x - 4) = 2x - 8\)
This inequality is equivalent to
\(3x - 2x > -8 - 5\)
or
\(x > -13\)

However, in our case:
\(x < 4\)
We find that the numbers x such that
\(x < 4\)
and
\(x > -13\)
are precisely those satisfying the inequality. This achieves what we wanted to
do. Notice that the preceding two inequalities can be written in the form:
\(-13 < x < 4\)

The set of numbers x satisfying such inequalities is called an interval. The
numbers -13 and 4 are called the endpoints of the interval. 

The set of numbers x such that 
\(3 < x < 7\) is an interval.

The set of numbers x such that:
\(3 \leq x \leq 7\)
is also called an interval. In this case we include the endpoints, 3 and 7, in
the interval. The word interval applies to both cases, whether or not we admit
the endpoints.

In general, let a,b be numbers with
\(a \leq b\)
Any one of the following sets of numbers is called an interval.

The set of numbers x such that
\(a < x < b\)
is called an open interval.

The set of numbers x such that 
\(a \leq x \leq b\)
is called a closed interval.

The set of numbers x such that
\(a \leq x < b\)
and
The set of numbers x such that
\(a < x \leq b\)
are both half open or half closed.

Again, by convention, it is customary to say that the set of all numbers x such
that 
\(x > 7\)
is an infinite interval. Similarly, the set of all numbers x such that
\(x < -3\)
is an infinite interval, and so is the set of numbers x such that
\(x < a\)
Again by convention, we may wish to include the endpoint. For instance, the set
of numbers x such that
\(x \geq 7\)
is also called an infinite interval. The set of numbers x such that
\(x \leq -3\)
is called an infinite interval.