Solving linear inequalities
is very similar to solving linear equations, except for one detail: you
flip the inequality sign whenever you multiply or divide the inequality
by a negative. The easiest way to show this is with some examples:
1)
Graphically,
the solution is:
The
only difference here is that you have a "less than"
sign, instead of an "equals" sign. Note that the solution
to a "less than, but not equal to" inequality
is graphed with a parentheses (or else an open dot) at the endpoint.
2)
Graphically,
the solution is:
Note
that "x"
does not have to be on the left, but it is often easier to picture
how to deal with it this way. Don't be afraid to rearrange things
to suit your taste.
3)
Graphically,
the solution is:
Same
ol', same ol', but with a "less than or equal to"
sign, instead of a plain "equals". Note that the solution
to a "less than or equal to" inequality is graphed
with a bracket (or else a closed dot) at the endpoint.
This
is the special case noted before. When we divided by the
negative
two, we had to flip the inequality sign!
The rule for number 5
above often seems unreasonable to students the first time they see it.
But think about it with numbers in there, instead of variables.
If 4 >
2 (which it is), then,
multiplying through by –1,
we get –4
< –2 (which is also
true):
If we hadn't flipped the
inequality, we would have ended up with "–4
> –2", which
clearly isn't true.
