1. A quadratic inequality can be written in the
form ax2+bx+c<0, ax2+bx+c>0, ax2+bx+c≤0, or ax2+bx+c≥0, where a, b, and c
are all real numbers and . a≠0
2. A solution to an inequality is an interval or
intervals on the x-axis with the property that all x values in the interval
satisfy the inequality.
3. Interval Notation:
(a, b) means all values between a and b not including a or b.
(a, b] means all values between a and b not including a but including b.
[a, b) means all values between a and b not including b but including a.
[a, b] means all values between a and b including a and b.
Note that "(" or ")" means that the number next to the
parenthesis are not included in the interval and "[" or "]" means that the
number next to the bracket is included in the interval.
Note that the smaller number is always listed on the left
in the interval notation.
(a,∞) means all values greater than a not including a.
[a,∞) means all values greater than a and including a.
(−∞,a) means all values less than a not including a.
(−∞,a] means all values less than a and including a.
Note that ∞ is always on the right fol lowed by a
parenthesis , and −∞ is always on the left preceded by a parenthesis .
4. Recall that linear inequalities are solved
algebraically almost like linear equations . The only difference is that anytime
you multiply or divide by a negative number, you must change the direction of
the inequality.
5. However, quadratic inequalities are not solved
algebraically like quadratic equations.
6. Solve Quadratic Inequalities by Graphing: To
solve ax2+bx+c<0, ax2+bx+c>0, ax2+bx+c≤0, or ax2+bx+c≥0, where a, b,
and c are all real numbers and a≠0 use the following steps.
a. Be sure that one side of the inequality is zero.
b. Enter the non- zero side of the inequality in the y =
key of the calculator.
c. Note that the function
can change signs only at x intercepts. Recall that the x- intercepts are
solutions to ax2+bx+c = 0, and these are the points where the graph crosses the
x-axis.
d. To find the x-intercepts in the calculator from the
graph of y = ax2+bx+c, use the command #2 zero under 2nd TRACE or CALC.
e. Solutions:
The solution to ax 2+bx+c>0 are all x values where the
graph is above the x-axis. (This is where you get y values > 0.)
The solution to ax2+bx+c≥0 are all x values where the
graph is above the x-axis and the solutions to ax2+bx+c=0. (Thus, the endpoints
of the interval are included in the solution set, so use the "]" or "["
notation.)
The solution to ax2+bx+c<0 are all x values where the
graph is below the x-axis. (This is where you get y values < 0.)
The solution to ax2+bx+c≤0 are all x values where the
graph is below the x-axis and the solutions to ax2+bx+c=0. (Thus, the endpoints
of the interval are included in the solution set, so use the "]" or "["
notation.)
7. Solve Quadratic Inequalities Algebraically Using
Test Numbers: See page 804.
Basic Idea: This method uses the same ideas we used
with the graphical method but without graphing. Find the point where the graph
can change signs (the x-intercepts). These points divide the x -axis into
intervals. Check which interval or intervals have the correct sign to solve the
inequality.
a. Be sure that one side of the inequality is zero.
b. Write the inequality as an equation (replace the
inequality sign with an equal sign) and solve it.
Note that the quadratic function
will have the same sign at all x values between these solutions.
(Only use real solutions. Complex solutions mean that the
function does not change signs, and so either all real numbers will be solutions
or none will be solutions.)
c. Put the solutions found above on a number, dividing the
number line into intervals.
d. Pick a test number in each interval. (These test
numbers can be any number in the interior of the interval.) Plug these numbers
into the quadratic function
and note the sign on the value obtained. On each interval, label the number line
with the sign found for that interval.
e. Write the solution in interval notation.
If you want > 0, take the intervals with +'s and use "("
and ")".
If you want ≥ 0, take the intervals with +'s and use "[" and "]".
If you want < 0, take the intervals with –'s and use "(" and ")".
If you want ≤ 0, take the intervals with –'s and use "[" and "]".
(For intervals that extent left or right indefinitely,
recall that ∞ is goes on the right followed by a parenthesis, and −∞ is goes on
the left preceded by a parenthesis.)
8. Note that this method can be extended to any polynomial
inequality or rational inequality .
