Quadratic Inequalities in 1 Variable
EXAMPLE 1: x2 - 2x – 15 ≥ 0
1. First, make sure the inequality is less than, less than or equal to,
greater than, or greater than or equal to zero. If it’s not, subtract
from both sides until it is.
2. Factor the inequality like it was an equation. Then, solve for x.
(x + 3)(x – 5) = 0
x = -3 or x = 5
3. After you get the two roots of the inequality, graph them on a number
line.
x = -3 or x = 5
4. Now you have to figure out which part of the number line to shade in.
Pick a test point on the number line that is not either of the roots and
substitute it into the original inequality. Zero is a good test point to
use.
0
x2 – 2x – 15 ≥ 0
(0)2 – 2(0) – 15 ≥ 0
0 – 0 – 15 ≥ 0
-15 ≥ 0 FALSE
5. If the inequality ends up false after you substitute the test point in,
like this one, you have to draw arrows coming from the roots that do
not include the part of the number line where the test point is. In this
case, you would draw arrows going in opposite directions from each
root. This means that x is greater than the bigger root and x is
less than the smaller root.
x ≤ -3 or x ≥ 5
If the inequality ends up true, then you draw a line between the two
roots that connects them. This would mean that x is greater than
the smaller root and less than the bigger root.
EXAMPLE 2: x2 - 8x + 12 ≤ 0
(x – 2)(x – 6)
x = 2 or x = 6
4
(4)2 – 8(4) + 12 ≤ 0
16 – 32 + 12 ≤ 0
-4 ≤ 0 TRUE
2 ≤ x ≤ 6
NOTE: If the sign in the inequality is less than or equal to, or greater
than or equal to (≤ ≥), the points on the number line are filled in and
the lines of the arrows are solid. If the sign in the inequality is less
than or greater than (< >), the points on the number line are open and
the lines of the arrows are dotted or dashed.
Quadratic Inequalities in 2 Variables
EXAMPLE 1: y ≥ (x-2)2 + 1
1. First graph the inequality on a graph like it was a quadratic equation.
You could also use a graphing calculator for this step.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
2. Now, because this is an inequality and not an equation, you have to
figure out which part of the graph to shade in. Pick a test point that
the graph does not cross and substitute into the original inequality. A
good test point to use is (0,0).
0
(0) ≥ ((0)-2)2 + 1
0 ≥ (-2)2 + 1
0≥4+1
0 ≥ 5 FALSE
3. If the inequality ends up false after you substitute the test point in,
you have to shade in the region of the graph that does not include the
test point.
If the inequality ends up true after you substitute the test point in,
you have to shade in the region of the graph that does include the
test point.
EXAMPLE 2: y < x2 – 3x + 2
0
(0) < (0)2 - 3(0) + 2
0<0+0+2
0 < 2 TRUE
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
NOTE: If the sign in the inequality is less than or equal to, or greater
than or equal to (≤ ≥), the points on the graph are filled in and the
parabola is solid. If the sign in the inequality is less than or greater
than (< >), the points on the graph are open and the parabola is dotted
or dashed.