Quadratic and Rational Inequalities
A quadratic inequality is any
inequality that can be put in
one of the forms
where a, b, and c are real numbers and a≠0.
Solving Quadratic Inequalities
Step 1: write the inequality in standard
form.
Step 2: solve the related quadratic equation.
Step 3: locate the boundary points on a
number line.
Step 4: construct a sign chart.
Step 5: the solution set is the interval that
produced a true statement.
Example 1: Solve a Quadratic
Inequality
Solve:
Solution:
Step 1: Write the inequality in
standard form.
Step 2: Solve the related quadratic equation.
(x1)(x3)=0
x+1=0 or x3=0
x=1 or x= 3
The boundary points are 1 and 3.
These two points divide the number
line into three test intervals, namely
(−∞,1) ,(−1,3) ,and (3,∞) .
Take a test point within each
interval and check the sign.
Test
Interval 
Test
Point 
X+1 
X3 
(x+1)(x3) 
(−∞,1) 
2 
 
 
+> 0 
(−1,3) 
0 
+ 
+ 
< 0 
(3,∞) 
4 
+ 
+ 
+>0 
The question is "where
?"
Our table shows that
in the interval (1,3).
So the solution set is
the interval (1,3)
Practice Exercises
Solving Rational Inequalities
Solve:
It is incorrect to multiply both sides by
x2 to clear fractions. The problem is
that x2 contains a variable and can be
positive or negative, depending on the
value of x. Thus, we do not know
whether or not to reverse the sense of
inequality.
Example 2:
solution: We begin by finding values
of that make the numerator and
denominator 0
Set the numerator and denominator equal to 0.
Solve
x=5 and x=2
The boundary points are 5 and 2.
Locate boundary points 5 and 2
on a number line.
These boundary points divide the
number line into three intervals,
namely (−∞,5),(5,2), and (2,∞).
Now, construct a sign chart: take
one test point from each interval
and check the signs.
Intervals 
Test
Points 
x+5 
x2 

(−∞,5) 
6 
 
 
+> 0 
(5,2) 
0 
+ 
 
< 0 
(2,∞) 
3 
+ 
+ 
+>0 
The given question is "for what values
of the quantity is
positive?"
Our table shows that
for
(−∞,5) or (2,∞).
Thus the solution set is
(−∞,5) or (2,∞).
The graph of the solution set on a
number line is shown as follows:
Practice Exercises
Solve: 
Answers: 

(5,2)
(−∞,3) or [4,∞)
[4,2] 
