

Quadratic
inequalities

Solving
quadratic inequalities 
Graphic solution of
quadratic inequalities 





Quadratic
inequalities

To solve a quadratic inequality we can examine the sign of the
equivalent quadratic function. 
The
xintercepts
or roots are the points where a quadratic function changes the sign. The
xintercepts determine
the three intervals on the xaxis in which the function is above or
under the xaxis, that is, where the function is positive or negative. 

Graphic solution of
quadratic inequalities 
Example:
Solve the inequality

x^{2}
+ 2x
+
3 ≤
0. 
Solution:
Solve the quadratic
equation ax^{2}
+ bx
+
c
=
0
to get the boundary points. 
The zeroes
or roots of equivalent function (see the graph
below) are the endpoints of the intervals and are included in the solution. 
The turning point V(x_{0},
y_{0}), 

The roots, 
x^{2}
+ 2x
+
3 =
0 



Solution: 




Example:
Solve the inequality 



Solution:
A fraction is negative if
the numerator and the denominator have opposite signs. 
Thus,
we have to solve the two systems of inequalities 
a) x^{ } 
2 <
0 and b)
x^{ }
2 >
0 
2x +
3 >
0 2x +
3 <
0 

x^{ }<
2 x^{ }>
2 
x^{ }>
3/2. x^{ }<
3/2 
Where b) represents the system
of the contradictory inequalities. 
The
red colored part of the graph of the
corresponding equilateral or rectangular
hyperbola shows the interval (region) where the
function's values are negative. 
Thus,
the solution is 3/2
<
x^{ }<
2. 



Or,
we can transform the given fraction to the quadratic inequality
by multiplying it by (2x
+
3)^{2}. 
So,
obtained is 
(x^{ }
2)(2x
+
3)
<
0
or
2x^{2}

x

6 <
0. 
The
roots of the corresponding function, 
x^{ }
2 = 0,
x_{1}
= 2
and 2x
+
3 = 0,
x_{2}
= 3/2 
Therefore,
the solution is 
3/2
<
x^{ }<
2. 
The
right figure shows that the graph of the
function is negative in this interval. 












Precalculus
contents B 



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