Inequalities Part I

Equalities are equations with an equal sign in the middle of two expressions. The equal sign says the expression on the left is equal to the expression on the right. Inequalities are connected by something other than an equal sign such as:


greater than>
greater than equal to
less than
less than equal to
not equal

Numbers may placed on both sides of an inequality which is a little boring, but we can also put algebraic expressions which can express a lot of meaning with just one inequality sign and a variable. Take for instance the sentence


x 5.

This one sentence says 6 is greater than 5. 7 is greater than 5. 8 is greater than 5, and so on to infinity.

Sometimes when working with numbers we may get a more complex algrebraic expression like:

.

In order to solve this inequality we must isolate x. Dividing both sides by 5 gives us the expression:
.

Now it is readily apparant that x is greater than or equal to 2.


A more complicated inequality will take more work.



This seems a bit complicated but with some manipulation we will be able to solve for x. First notice the numerator. It is a quadratic expression that factors and cancels like this:


Now we are left with the remaining terms:


Next notice that the inequality is undefined at the points 3 and 6 because putting either of those values makes the denominator 0, and dividing by 0 is not defined. Lets see what happens around those values and also -5 since it makes the entire expression 0. Lets substitute for x the value -10. That is just left of -5. There was no particular reason we chose -10. Any number less than -5 would work. -10 makes the numerator negative. -10 also makes (x-3) and (x-6) negative.

Whenever we multiply or divide any even number of negatives we get a positive number. Whenever we multiply or divide any odd number of negative signs we get a negative number. -10 gives us 3 negatives. So negative -10 makes the expresion negative which makes the entire inequality true. It doesn't just hold for -10. Any number less than or equal to -5 will make the entire inequality true. We included -5 because the inequality is a less or equal to.

What about numbers between -5 and 3. Lets try substituting 0 which is between -5 and 3. 0 makes the numerator positive and gives us two negatives in the denominator for a total of two negatives. Two negative numbers multiplied with a positive number is a positive number. Since the inequality is false for 0 it is also false for all the numbers between -5 and 3. This time we are not worried about how the inequality behaves at exactly 3 because 3 makes the expression undefined.

The next interval is the numbers between 3 and 6. Substituting 4 for x gives us two positives and a negative. This is an odd number of negatives so the expression is less than 0 for all the numbers between 3 and 6.

Now for the last interval. 10 is to the right of 6 on the number line. Substituting 10 gives us three positives which is also a positive. This makes the entire inequality false.

Lets summarize all the numbers which makes this inequality true. All the numbers less or equal to -5 makes the inequality true, and is symbolized by (- ,- 5]. All the numbers between 3 and 6 make the inequality less than 0 which makes it true and is symbolized (3,6). Using set notation we make our final answer
(- , - 5] (3,6). We can check our answer with a graph: