2. Learning Objectives
At the end of this unit, you should be able to:
identify
quadratic
inequalities;
01
solve
problems
involving
quadratic
inequalities
03
solve
quadratic
inequalities;
and
02
9TH GRADE
4. A quadratic
inequality (in
one variable) is
an inequality
with one
unknown
(usually ) of
degree 2.
Simply put, it is
like a quadratic
equation but the
equal sign is
replaced by a
less than, greater
than, at most, or
at least symbol.
9TH GRADE
5. It can be written in the following forms:
𝑎𝑥2
+ 𝑏𝑥 + 𝑐 > 𝟎
9TH GRADE
where and are
real numbers,
and a ≠ 0
𝒂𝒙𝟐
+ 𝒃𝒙 + 𝒄 < 𝟎
𝒂𝒙𝟐
+ 𝒃𝒙 + 𝒄 ≥ 𝟎
𝒂𝒙𝟐
+ 𝒃𝒙 + 𝒄 ≤ 𝟎
7. If the simplified
form of an
inequality
doesn’t have 2 as
the highest
power of the
variable x, it is
not a quadratic
inequality.
Things to remember about quadratic
inequalities:
When
simplified,
quadratic
inequalities
retain 2 as the
highest power
of the variable
x.
9TH GRADE
The difference
between a
quadratic
equation and a
quadratic
inequality is that
the equation is
equal to some
number while
inequality is
either less than
or greater than
11. Solve the quadratic inequality 𝒙𝟐 + 𝒙 > 𝟔
Step 1: Write one side of the inequality
in the form and the other side zero.
𝑥2
+ 𝑥 > 6
𝑥2
+ 𝑥 − 6 > 0
12. Step 2: Treat the inequality as a
quadratic equation. Solve the
equation. The solutions are the critical
numbers of the inequality.
𝑥2
+ 𝑥 − 𝑏 = 0
𝑥 + 3 𝑥 − 2 = 0
13. Step 3: Use the critical
numbers to determine
the test intervals. The
test intervals are 𝑥 <
− 3, −3 < 𝑥 < 2, 𝑥 > 2
14. Step 4: Choose one test number in each
test interval and evaluate the inequality at
that value. You may use the original
inequality or factored form of the inequality
when evaluating.
Original Inequality: 𝑥2
+ 𝑥 > 6
Factored Inequality: 𝑥 + 3 𝑥 − 2 > 0