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.
