To this point in this chapter we’ve concentrated on solving equations.  It is now time to switch gears a little and start thinking about solving inequalities.  Before we get into solving inequalities we should go over a couple of the basics first.

At this stage of your mathematical career it is assumed that you know that

means that a is some number that is strictly less that b.  It is also assumed that you know that

means that a is some number that is either strictly bigger than b or is exactly equal to b.  Likewise it is assume that you know how to deal with the remaining two inequalities. > (greater than) and  (less than or equal to).

What we want to discuss is some notational issues and some subtleties that sometimes get students when the really start working with inequalities. 

First, remember that when we say that a is less than b we mean that a is to the left of b on a number line.  So,

is a true inequality.

Next, don’t forget how to correctly interpret  and .  Both of the following are true inequalities.

In the first case 4 is equal to 4 and so it is “less than or equal” to 4.  In the second case -6 is strictly less than 4 and so it is “less than or equal” to 4.  The most common mistake is to decide that the first inequality is not a true inequality.  Also be careful to not take this interpretation and translate it to < and/or >.  For instance,

is not a true inequality since 4 is equal to 4 and not strictly less than 4.

Finally, we will be seeing many double inequalities throughout this section and later sections so we can’t forget about those.  The following is a double inequality.

In a double inequality we are saying that both inequalities must be simultaneously true.  In this case 5 is definitely greater than -9 and at the same time is less than or equal to 6.  Therefore, this double inequality is a true inequality.

On the other hand,

is not a true inequality.  While it is true that 5 is less than 20 (so the second inequality is true) it is not true that 5 is greater than or equal to 10 (so the first inequality is not true).  If even one of the inequalities in a double inequality is not true then the whole inequality is not true.  This point is more important than you might realize at this point.  In a later section we will run across situations where many students try to combine two inequalities into a double inequality that simply can’t be combined, so be careful.

The next topic that we need to discuss is the idea of interval notation.  Interval notation is some very nice shorthand for inequalities and will be used extensively in the next few sections of this chapter.

The best way to define interval notation is the following table.  There are three columns to the table.  Each row contains an inequality, a graph representing the inequality and finally the interval notation for the given inequality.

Inequality	Graph	Interval Notation

Remember that a bracket, “[” or “]”, means that we include the endpoint while a parenthesis, “(” or “)”, means we don’t include the endpoint.

Now, with the first four inequalities in the table the interval notation is really nothing more than the graph without the number line on it.  With the final four inequalities the interval notation is almost the graph, except we need to add in an appropriate infinity to make sure we get the correct portion of the number line.  Also note that infinities NEVER get a bracket.  They only get a parenthesis.

We need to give one final note on interval notation before moving on to solving inequalities.  Always remember that when we are writing down an interval notation for an inequality that the number on the left must be the smaller of the two.

It’s now time to start thinking about solving linear inequalities.  We will use the following set of facts in our solving of inequalities.  Note that the facts are given for <.  We can however, write down an equivalent set of facts for the remaining three inequalities.

1. If
   
    then
   
    and
   
    for any number c.  In other words, we can add or subtract a number to both sides of the inequality and we don’t change the inequality itself.
   
   
2. If
   
    and
   
    then
   
    and
   
   .  So, provided c is a positive number we can multiply or divide both sides of an inequality by the number without changing the inequality.
   
   
3. If
   
    and
   
    then
   
    and
   
   .  In this case, unlike the previous fact, if c is negative we need to flip the direction of the inequality when we multiply or divide both sides by the inequality by c. 

These are nearly the same facts that we used to solve linear equations.  The only real exception is the third fact.  This is the important fact as it is often the most misused and/or forgotten fact in solving inequalities.

If you aren’t sure that you believe that the sign of c matters for the second and third fact consider the following number example.

I hope that we would all agree that this is a true inequality.  Now multiply both sides by 2 and by -2.

Sure enough, when multiplying by a positive number the direction of the inequality remains the same, however when multiplying by a negative number the direction of the inequality does change.

Okay, let’s solve some inequalities.  We will start off with inequalities that only have a single inequality in them.  In other words, we’ll hold off on solving double inequalities for the next set of examples.

The thing that we’ve got to remember here is that we’re asking to determine all the values of the variable that we can substitute into the inequality and get a true inequality.  This means that our solutions will, in most cases, be inequalities themselves.