Professor Dave here, let’s talk about inequalities.
Algebra is all about equations.
One side equals the other side, and we move things around until we can solve for a variable.
But there is another type of construct in algebra that we will need to use, and it is
related to the equation.
It is called the inequality.
Instead of an equation that equates two terms, telling us that they are equal, an inequality
tells us how two terms are not equal, but rather that one is either greater than the
other or less than the other.
We can manipulate and solve inequalities exactly the same way that we manipulate equations,
with one or two tiny additional rules, so let’s take a look at these now.
First let’s examine something simple, like X is greater than three.
Unlike a simple equation, where X has precisely one value, X has many values, and any value
for X that satisfies the inequality is a solution for X. Four could work, or ten, or a billion,
or 3.00001.
Any number imaginable that is bigger than three is a solution to this inequality, and
this can be represented on a number line using an open circle, implying that three is not
part of the solution.
Anything less than or equal to three, like zero, or negative three, or 2.999, or three
itself, will not be a solution.
We will see inequalities set up like some of the equations we’ve seen as well.
How about X plus two is less than seven.
As you might expect, all we do is subtract two from both sides, which gives us X is less
than five, so any value less than five is a solution, while five or above is not.
We will also see these symbols, which mean greater than or equal to, and less than or
equal to.
These work the same way as the other symbols except that they include the specific value
shown rather than excluding it.
If we have three X plus four is greater than or equal to sixteen, we subtract four from
both sides, then divide by three, and we get X is greater than or equal to four.
Now on the number line, we represent this with a closed circle, meaning that it is filled
in rather than empty, to represent that the number four is included in this solution.
So everything here works the same way it does with equations, but there is one difference
we need to point out.
If we are multiplying or dividing by a negative number, we need to reverse the direction of
the inequality symbol.
For example, if negative X is greater than six, we would either multiply or divide both
sides by negative one to get X, but if we do that, we have to switch the greater than
to a less than.
X is less than negative six.
To see why this is the case, let’s test this solution.
Take the first version and plug in negative ten.
The negative version of negative ten, or the opposite of negative ten, is ten, and that
is indeed greater than six.
Now with our revised version, negative ten is also less than negative six, so we can
see that this flipped version is indeed the correct counterpart.
One way of rationalizing this is by looking at a number line.
First let’s remember that negative numbers closer to zero are larger than negative numbers
farther away from zero, so although it may for a moment seem like negative ten is bigger
than negative five, it’s actually the other way around.
Then we must understand that when we multiply and divide these values by positive numbers,
we are just stretching and shrinking the line as it is.
But if we multiply or divide by a negative number, we are inverting the sign for every
value, which makes it look like the mirror image of the number line.
To compensate for this, it makes sense to flip the whole thing over, which is essentially
what is achieved by flipping the sign of the inequality.
We can verify this by using whole numbers.
One is less than two, but if we multiply by negative one, we would get negative one is
less than negative two, which is not true.
The only way this remains accurate is by flipping the symbol to get negative one is greater
than negative two.
Other than this little rule, which can sometimes be tricky to remember, solving simple inequalities
like this is no different than solving simple equations, so let’s check comprehension.