4.9 Graph and Solve
Quadratic Inequalities
Quadratic inequality
in two variables
An inequality that can be written in the form y<
ax2 + bx + c, y £ ax2
+ bx+ c,
y > ax2 + bx+ c, or
y ³ ax2+
bx + c
Quadratic inequality
in one variable
An inequality that can be written in the form ax2
+ bx+ c < 0, ax2 + bx+ c £
0,
ax2 + bx+ c > 0, or ax2+ bx+ c
³
0
GRAPHING A QUADRATIC INEQUALITY IN TWO VARIABLES
To graph a quadratic
inequality, follow these steps:
Step
1 Graph the parabola with equation
y
= ax2+ bx + c.
Make
the parabola _dashed_ for inequalities with < or > and _solid_
for inequalities with £ or ³.
Step 2 Test
a point (x, y) _inside_ the parabola to determine whether
the point is a solution of the inequality.
Step
3 Shade the region _inside_ the parabola if the
point from Step 2 is a solution. Shade the region _outside_ the parabola
if it is not a solution.
Example 1
Graph
a quadratic inequality
Graph y £
- x2 +
2x +
3.
1.
Graph
y = -x2
+ 2x + 3. The inequality symbol is £,
so make the parabola _solid_ .
2.
Test
the point (0, 0).
_0
£ -(0)2
+ 2(0) + 3_
_0
£ 3_
So,
(0, 0) _is a solution_ .
3.
Shade
the region _inside_ the parabola.
Graph y > x2
- 2.
1.
Example 2
Graph
a system of quadratic inequalities
Graph the system of quadratic inequalities.
y
> x2 - 2 Inequality
1
y
£
-x2
-
3x + 4 Inequality
2
Solution
1.
Graph y > x2
- 2. The graph is the region _inside_ (but
not including) the parabola
y = _x2 -
2_.
2.
Graph y £
-x2
-
3x + 4. The graph is the region _inside_ and including the
parabola
y = -x2
- 3x
+ 4.
3.
Identify the region
where the two graphs overlap. This region is the graph of the system.
Graph the system.
2.
y
< -x2
+ 3
y
³
2x2 + 3x - 2
Example
3
Solve
a quadratic inequality using a table
Solve x2 +
3x £
4.
Rewrite the inequality as x2 +
3x - 4 £ 0. Then make a
table of values.
Notice that x2 + 3x -
4 £
0 when the values of x are between _-4_
and _1_, inclusive.
The solution of the inequality is _-4
£ x £
1_.
Example 4
Solve a quadratic inequality by graphing
Solve
-3x2 - 5x + 3 £ 0.
The
solution consists of the x-values for which the graph of y = -3x2 - 5x + 3 lies
_on or below_ the x-axis. Find the graph's x-intercepts by
letting y = 0 and using
_the quadratic formula_ to solve for x.
0
= -3x2 - 5x + 3
x »
_0.47_ or x » _-2.14_
Sketch
a parabola that opens _down_ and has _0.47_ and _-2.14_ as x-intercepts.
The graph lies _on or below_ the x-axis to the left of (and
including) x = _-2.14_ and to the right of
(and including) x = _0.47_.
The
solution is approximately _x £ -2.14 or x ³ 0.47_.
Example
5
Solve
a quadratic inequality algebraically
Solve x2
+ x ³
12.
First, write and solve the equation obtained by
replacing ³ with _=_.
_x2
+ x = 12_ Write corresponding equation.
_x2
+ x -
12 = 0_ Write in standard form.
_(x
+ 4)(x -
3) = 0_ Factor.
_x
= -4 or x = 3_ Zero product property.
The numbers _-4
and 3_ are the critical x-values of the
inequality x2 + x ³ 12. Plot
_-4
and 3_ on a number line, using _solid_ dots. The critical
x-values partition the number line into three intervals. Test an x-value
in each interval to see if it satisfies the inequality
Test x = _-6_: Test x = 0 :
_(-6)2
+ (-6) = 30 ³
12_ _(0)2
+ (0) = 0 ≱ 12_
Test x = _5_:
_(5)2
+ (5) = 30 ³ 12_
The solution is _x £
-4 or x ³
3_.