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.
