Class 2 -- Inequalities

I. Simple Inequalities
II. Graphing Inequalities
III. Compound Inequalities
IV. Inequality Word Problems

(NOTE: This is a short lecture)

Last week we learned how to solve algebraic equations in one
variable.

This week we're going to extend that knowledge to solving inequalities 
rather than equalities.

Fortunately, almost everything we learned in solving equalities will
carry over -- there are just a couple of twists.

An inequality is just like an equation, except that we replace the =
sign with an inequality sign: <, <=, >, >=

Just to remind you what these signs mean:

(on board)
<	Less than
<=	Less than or equal to
>	Greater than
>=	Greater than or equal to

A useful mnemonic for this is that the sign is the mouth of a fish,
and the fish opens its mouth for the bigger morsel. (Draw picture)


I. Simple Inequalities

Let's consider a simple equation we know how to solve, and replace the 
= sign with an inequality:

(on board)
3x - 9 < 15

The goal is to get the variable by itself, just as we do with an
equation.

OK, how would we solve this if it were an equation? (Let class do
this.)

In the end we get: x < 8 (on board)

That's all there is to it: just like an equation.

Let's consider another example:

5 - 2x >= 7

How could we solve this? (Let class do it -- they'll pick one of the
two ways below)

Note that we can approach this two different ways. One way would be to 
move the x to the right side:

(on board)
5 - 2x >= 7
   +2x   +2x

5 >= 7 + 2x
-7  -7

-2 >= 2x
-1 >= x

So, in the end we get x <= -1.

Now we could also solve the problem this way:

(on board)
5 - 2x >= 7
-5       -5

-2x >= 2

OK, so now we divide by -2. Note that if we do this and leave the
inequality as it is, we'll get

x >= -1.

But something is wrong here! That's exactly the opposite of what we
got when we did the problem the other way.

The problem is that when we multiply or divide by a negative, we have
to flip the inequality: > becomes <, >= becomes <=, and vice-versa.

Thus, when we divide by -2, we should write

(on board)
-2x / -2 <= 2 / -2
x <= -1

Now we get agreement.

That's the only difference between solving inequalities and equations: 
with inequalities, you have to flip the inequality when you multiply
or divide both sides by a negative.


II. Graphing Inequalities

(NOTE: the book uses different notation for including or not including
a point on a number line.  e.g. the book uses ( and [ rather than an open
and closed circle.  Mention this, and mention that we don't care what 
convention they use)

To help get a handle on what we're talking about when we write an
inequality, it's often helpful to graph the inequality.

By graphing an inequality, I mean that we draw a number line and
darken in the parts that correspond to the inequality.

For example, consider x <= -1, the inequality we just got.

To graph this, we first draw a number line.

(draw number line)

Then we darken in all the points less than -1.

(do this)

We then put a filled cirlce on -1 itself, to indicate that -1 is
included in the set.

(do this)

OK, let's do another:

(on board)
Graph x > 1.

So we draw the number line and darken the part to the right of 1.

(draw on board)

We draw an unfilled circle over 1 itself to indicate that it is not
included in the set.

(do this)

That's all there is to it.


III. Compound Inequalities

Sometimes we see two inequalities put together -- this is called a
compound inequality.

Consider this:

(on board)
3x + 5 > 2   and   4 - 2x > -2

Here we're being told two statements about x. To solve in this
situation, we begin by considering and solving each one separately.

Let's start with 3x + 5 > 2. How would we solve this? (Let class do
it.)

3x + 5 > 2
    -5  -5
3x > -3
x > -1

OK, now let's solve the other one. (Let class do this.)

4 - 2x > -2
-4       -4
-2x > -6
x < 3

So now we have:

(on board)
x > -1 and x < 3

Clearly this is telling us that we want all the numbers between -1 and 
3.

We can combine these statements to write them in a somewhat simpler
fashion:

(on board)
-1 < x < 3

We can also see this graphically: let's graph x > -1 and x < 3

(draw number lines on board)

Note that we want "and", meaning that we want all the numbers that are
included in both shaded region.

In other words, we want the region where the two shaded regions
intersect one another.

If we shade in that region, we see:

(draw number line on board)

Clearly the picture we've just drawn corresponds to -1 < x < 3.

Note that is is possible in a given problem that there is no
overlap. In that case, the problem simply has no solution.

To represent that, we write that the solution is (empty set symbol on
baord).

This symbol is called the empty set -- it's a way of saying that the
set of numbers that solve these inequalities is empty.

OK, let's try another example:

3x + 1 <= -2   or   3 - 2x <= 1

Again, we solve these one at a time. (Let class do this.)

(on board)
3x + 1 <= -2
3x <= -3
x <= -1

(on board)
3 - 2x <= 1
-2x <= -2
x >= 1

There's no way to write this in a simpler fashion than it already is:
x<-1 or x>1.

Let's graph this. First we'll graph each of the two inequalities
independently.

(draw number lines)

Now we'll combine them. Note that here we have "or" instead of
"and".

If we have "and", that means we want all the points that lie in both
regions, and only those points.

If we have "or", that means we want all the points that lie in either
of the two shaded regions. Thus, the graph looks like:

(draw number line)

The technical term for the combination with or is taking the "union",
since we are combining the two sets.

Thus, "and" means intersection, and "or" means union.


IV. Inequality Word Problems

Just as it is possible to translate equations from algebra into
English and back, the same is true for inequalities.

Fortunately, translating inequalities is just like translating
equations, except that we add some new key phrases: less than, greater 
than, etc.

For example, how could we translate this into English:

(on board)
x >= 6

(Let class answer)

OK, let's try going the other way. Turn this into algebra:

(on board)
Meat must be cooked to an internal temperature of at least 180F to
ensure that it is safe.

Can we write an algebraic expression for this sentence? (Let class
answer).

As you can see, it's just like equations.

Suppose I added on to what I have written:

(on board)
A chef cooking hamburgers generally finds that the internal
temperature is 20% cooler than the external temperature. What is the
minimum external temperature the hamburger should reach?

Can we solve this? (let class solve)

0.8 T > 180
T > 180 / 0.8
T > 225

So the answer is:

(on board)
The external temperature should be at least 225F.

Another common type of inequality word problem involves tolerances.

With a tolerance, we specify a range that something can fall within,
either as an absolute value or as a percentage.

For example:

(on board)
A lathe operator is making a brake part that must be 12 cm wide with
a tolerance of 0.01 cm. Write a compound inequality expressing the
possible range in widths.

Let's take this problem step by step. First, can we write an
expression for the minimum width.

w > 11.99

And similarly for the maximum:

w < 12.01

So the compound is 11.99 < w < 12.01.

Of course this is more interesting if we add some complexity:

A goldsmith is making 10 grams of a new alloy by mixing pure (100%)
gold with a 50% gold, 50% copper mixture. If she wants the new alloy
to be 60-80% gold, how much pure gold can she use?

Solution:

Let x = amount of pure gold, 10 - x = amount of 50/50 alloy.

The amount of gold in the mixture is x + 0.5(10 - x) = 5 + 0.5 x.

We want this to be between 6 and 8 grams so,

6 < 5 + 0.5x < 8
1 < 0.5x < 3
2 < x < 6

She can use 2-6 grams of pure gold.