Solving linear inequalities
is very similar to solving
linear equations,
except for one small but important 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 between the linear equation "x + 3 = 2"
and this linear inequality is
that I have a "less than" sign, instead of an "equals"
sign. The solution method is exactly the same: subtract 3 from
either side.
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,
indicating that the endpoint is not included within the solution.
2)
Graphically,
the solution is:
The
only difference between the linear equation "2
− x = 0" and
this linear inequality is the "greater than" sign
in place of an "equals" sign.
Note that
"x"
in the solution does not "have" to be on the left.
However, it is often easier to picture what the solution means
with the variable on the left. Don't be afraid to rearrange
things to suit your taste.
3)
Graphically,
the solution is:
The
only difference between the linear equation "4x + 6 = 3x − 5"
and this inequality is the "less than or equal to"
sign in place of a plain "equals" sign. The solution
method is exactly the same.
Note that the
solution to a "less than or equal to" inequality is
graphed with a square bracket (or else a closed dot) at the
endpoint, indicating that the endpoint is included within the
solution.
4)
Graphically,
the solution is:
The
solution method here is to divide both sides by a positive two.
This
is the special case noted above. When I divided by the negative two, I had to flip the inequality sign.
The rule for example 5 above often seems unreasonable to students the first time they see it.
But think about inequalities with numbers in there, instead of variables.
You know that the number four is larger than the number two: 4
> 2. Multiplying
through this inequality by −1,
we get −4
< −2, which the
number line shows is true:
If we hadn't flipped the
inequality, we would have ended up with "−4
> −2", which
clearly isn't true.
