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Quadratic Inequalities
IES Sierra Nevada
Algebra
Quadratics
Before we get started let’s review. A quadratic equation is an equation that canbe written in the form , where a, b and c are real numbers and a cannot
equalzero.
In this lesson we are going to discuss quadraticinequalities.
02 cbxax
Quadratic Inequalities
What do they look like? Here are some examples:
0732 xx
0443 2 xx
162 x
Quadratic Inequalities
When solving inequalities we are trying to find all possible values of the variablewhich will make the inequality true.
Consider the inequality
We are trying to find all the values of x for which the
quadratic is greater than zero.
062 xx
Solving a quadratic inequality
We can find the values where the quadratic equals zero
by solving the equation, 062 xx
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 Test Point
Evaluate in the inequality True/False
2,
3,2
,3
06639633 2
06600600 2
066416644 2
3x
0x
4x
True
True
False
062 xx
062 xx
062 xx
Solving a quadratic inequality
Thus the intervals make up the solution set for the quadratic inequality, .It’s representation is:
062 xx ,32,
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 First find the zeros by solving the equation,
Now consider the intervals around the solutions and test a value from each interval in the inequality.
0132 2 xx0132 2 xx
1or2
1 xx
Example 2:
Interval Test Point Evaluate in Inequality True/False
2
1,
1,
2
1
,1
0x
4
3x
2x
0110010302 2
08
11
4
9
8
91
4
33
4
32
2
0316812322 2
False
True
False
0132 2 xx
0132 2 xx
0132 2 xx
Example 2:
Thus the interval makes up the solution set for
the inequality .
Plot the solution!!
0132 2 xx
1,
2
1
Example 3:
Solve the inequality .
First find the solutions.
WHAT CAN WE DO NOW??
12 2 xx
012or12 22 xxxx
22
12411 2
x4
71
Practice Problems
02452 xx
012 2 xx
0116 2 x
0253 2 xx
0123 2 xx
06135 2 xx
09 2 x
0152 2 xx
452 xx
422 xx
