Quadratic Inequalities
IES Sierra Nevada
Algebra
Quadratics
Before we get started let’s review.
A quadratic equation is an equation that can
be written in the form ax2  bx  c  0 ,
where a, b and c are real numbers and a cannot equal
zero.
In this lesson we are going to discuss quadratic
inequalities.
Quadratic Inequalities
What do they look like?
Here are some examples:
x 2  3x  7  0
3x  4 x  4  0
2
x  16
2
Quadratic Inequalities
When solving inequalities we are trying to
find all possible values of the variable
which will make the inequality true.
Consider the inequality
x2  x  6  0
We are trying to find all the values of x for which the
quadratic is greater than zero.
Solving a quadratic inequality
We can find the values where the quadratic equals zero
by solving the equation, x  x  6  0
2
Solving a quadratic inequality
Now, put these values on a number line and we can
see three intervals that we will test in the inequality.
We will test one value from each interval.
Solving a quadratic inequality
Interval
 ,2
Test Point
Evaluate in the inequality
True/False
x2  x  6  0
x  3
 32   3  6  9  3  6  6  0 True
x2  x  6  0
 2, 3
 3, 
x0
02  0  6  0  0  6  6  0
False
x2  x  6  0
x4
42  4  6  16  4  6  6  0
True
Solving a quadratic inequality
Thus the intervals  ,2  3,  make up the
solution set for the quadratic inequality, x 2  x  6  0 .
It’s representation is:
Summary
In summary, the steps for solving quadratic
inequalities are:
1. Solve the equation.
2. Plot the solutions on a number line creating the intervals.
3. Pick a number from each interval and test it in the original inequality. If the result
is true, that interval is a solution to the inequality.
4. Write properly the solution (the interval and the representation)
Example 2:
Solve 2 x  3x  1  0
2
First find the zeros by solving the equation, 2 x  3x  1  0
2
1
x  or x  1
2
Now consider the intervals around the solutions and
test a value from each interval in the inequality.
Example 2:
Interval
Test Point
Evaluate in Inequality
True/False
2 x 2  3x  1  0
1

  , 
2

x0
20  30 1  0  0 1  1  0
2
False
2 x 2  3x  1  0
1 
 ,1
2 
9 9
1
 3  3
2   3   1    1   0
8 4
8
 4  4
2
3
x
4
True
2 x 2  3x  1  0
1,  
x2
22  32 1  8  6 1  3  0
2
False
Example 2:
Thus the interval
the inequality
1 
 ,1
2 
makes up the solution set for
2 x 2  3x  1  0 .
Plot the solution!!
Example 3:
Solve the inequality  2 x 2  x  1 .
First find the solutions.
 2x2  x  1 or  2x2  x 1  0
 1   1  4 2  1  1   7

x
4
2 2
2
WHAT CAN WE DO NOW??
Practice Problems
x 2  5 x  24  0
12  x  x 2  0
3x 2  5 x  2  0
5x 2  13x  6  0
9  x2  0
2 x 2  5x  1  0
16x 2  1  0
x 2  5 x  4
3x 2  2 x  1  0
x2  2x  4