Inequalities

As we start this section, you should recall the basic rules which involve order (these are BR10, BR11, and BR12), that were introduced in the Real Numbers section. (These should start becoming quite familiar to you now.)

We note the following:

    a < b implies that b > a implies that b-a > 0
and
    a <= b implies that b >= a implies that b-a > 0 or b = a.

Here are some rules for ordering real numbers. Try proving these yourself. If you run into difficulties, proofs or hints to the proofs are given for some.

Let a, b, c and d be real numbers:

    1)
    a < 0 and b < 0 implies that ab > 0. Hint

    2)
    a < b and b < c implies that a < c. Proof

    3)
    a < b implies that a + c < b + c. Hint

    4)
    a < b and c > 0 implies that ac < bc Hint

    5)
    a < b and c < 0 implies that ac > bc. Multiplying by a negative number reverses the inequality.

    6)
    a > 1 implies that a2 > a. Hint

    7)
    0 < a < 1 implies that a2 < a

    8)
    0 <= a < b implies that a2 < b2 Proof

    9)
    0 <= a, 0 <= b, and a2 < b2 implies that a < b. Proof

Now that we have all of these rules, we can start solving inequalities. This is done by manipulating the inequality into a form that has the variable on one side and has a real expression on the other side of the inequality. For instance, if the variable is x and the real expression is represented by a then the final form of the inequality is one of the following:

    x > a
    x >= a
    x < a
    x <= a
Let have a peek at a few examples.

When the inequalities involve absolte values, you must be very careful with the use of the words and. and or . They end up giving quite different results. Take a look here to see what we mean. When we have the absolute values, the and. condition applies when we have < or <= signs and the or. condition applies when we have > or >= signs. So we have:

    |x| < a
    means -a < x < a (which is the same as -a < x and x < a).
    |x|<= a
    means -a<= x<= a (which is the same as -a <= x and x <= a)
    |x| > a
    means x < -a or x > a
    |x| >=a
    means x >= -a or x >= a
Let's check out some examples of inequalities with absolute values.

We can also have inequalities which involve polynomials. To solve these, we usually manipulate the inequality so 0 is on one side of the inequality then we factor the other side. Here are some examples.


Introductory Exercises Moderate Exercises
menu