Solving Linear Inequalities
• An inequality is an equation with ≤,<,≥ or > instead of
an equal sign
• A linear inequality is an inequality that is linear in form (there are no
roots, higher powers,
rationals, etc)
• Solving inequalities is the same as solving regular equations, but you need to
be careful of the
direction of the inequality sign. Multiplying or dividing the equation by a
negative swaps the
direction of the inequality sign (i.e. > would become <)
• A compound inequality is formed by putting together two inequalities with the
word
− and (meaning intersection, both have to be satisfied to be a solution)
− or (meaning union, either has to be satisfied to be a solution)
• The compound inequality with and could be joined in one statement called a
conjunction in the
manner below
x > 5 and x ≤12
5 < x ≤12
• The compound inequality with or is called a disjunction and cannot be
rewritten in shorter form
Recap of the Properties:
• Adding any quantity to both sides won’t change the inequality
i.e. a < b and k is any real, =>a + k < b + k
• Multiplying both sides by a POSITIVE constant won’t change the inequality
i.e. a < b, k > 0=>ak < bk
• Multiplying both sides by a NEGATIVE constant WILL change the direction of the
inequality
i.e. a < b, k < 0=>ak > bk
• Taking the reciprocal WILL change the direction of the inequality
i.e.
Describing the Intervals:
• There are several ways to describe intervals
– interval notation uses parenthesis, “(” and brackets, “]”
– set notation uses inequality symbols, “>”
– graphing uses the number line with open and closed circles connected with
lines or shading
• An interval can be open (meaning you don’t actually “contain” the endpoint).
The open end is
described with “)” “>” and/or open circles
• Or an interval can be closed (meaning you do contain the endpoint). The closed
end is described
with “]” “ ≥ ” and/or closed circles
• Example. To describe the values for x greater than 5
(5,∞)
x > 5
Some Common Mistakes (with NON-linear equations):
• When solving for a product of two quantities, you must take into account
that a negative times a
negative is a positive.
i.e. to solve for (x − 3)(x + 2) < 0 , you can’t just take when both are
negative. You have to do
what is called a “sign diagram” This can be done in one of two ways…
1) Look at the sign of x – 3 and x + 2 separately on a chart
You can see that (x − 3)(x + 2) < 0 when –2 < x < 3
2) Still do a chart, but just “test” values to the left
and right of your “zeros”
To the left of –2 (say –4), (−4 − 3)(−4 + 2) > 0
Between –2 and 3 (say 0), (0 − 3)(0 + 2) < 0
To the right of 3 (say +4), (4 − 3)(4 + 2) > 0
Which gives the same result as above
• Don’t ever just divide by x if you’re trying to solve an
inequality for x. Not only are you neglecting
values, you may be inadvertently dividing by a negative.
i.e. x2 < x does not reduce to x <1.
Solving correctly you get x2 − x < 0, x(x −1) < 0 so (with sign diagram) 0 < x
< 1
• When solving inequalities that contain a squared term,
you can’t just take the square root as
normal. You have to include the +/- with the variable, NOT the constant term.
i.e. to solve
,
Some Exercise Problems:
• Example. Solve 5y −5+ y ≤ 2 − 6y −8
• Example. Solve (x +1)(x + 2) > x(x +1)
• Example. Solve −3 <1− 2x ≤ 3
• Example. Solve 5x +11≤ −4 or 5x +11≥ 4
Absolute Value with Inequalities:
•
in other words, − k ≤ x ≤ k
• Example. Solve
