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,
a <
b
means that a is some number that is strictly
less that b. It is also assumed that you know that,
a ≥
b
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,
– 1000 < 0
is a true inequality.
Next, don’t forget how to correctly interpret ≤ and ≥ .
Both of the following are true inequalities.
4 ≤ 4 –6 ≤ 4
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,
4 < 4
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.
–9 < 5 ≤ 6
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,
10 ≤ 5 < 20
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.
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 a < b then a + c < b + c
and a – c < b – c 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 a < b and c > 0 then ac < bc
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.
. 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 a < b and c < 0 then
and a b c c>
. 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.
and a b c c>
. 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.
– 3 < 5
I hope that we would all agree that this is a true
inequality. Now multiply both sides by 2 and by – 2.
– 3 < 5 – 3 < 5
– 3(2)
< 5(2)
– 3(– 2) > 5(– 2)
– 6 < 10 6 > – 10
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.
Now, let’s solve some double inequalities. The process
here is similar in some ways to solving single inequalities and yet very different in other ways. Since there are two
inequalities there isn’t any way to get the variables on “one side” of the inequality and the
numbers on the other.
When solving double inequalities make sure to pay
attention to the inequalities that are in the original problem. One of the more common mistakes here is to start with a
problem in which one of the inequalities is < or > and the other is ≤ or ≥
, as we had in the first two parts of the previous example, and then by the final answer they are both < or > or
they are both ≤ or ≥ . In other words, it is easy to all of a sudden make both of the inequalities the
same. Be careful with this.
Sumber
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