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4.9 Graph and Solve Quadratic Inequalities
Goal  Graph and solve quadratic inequalities.
Your Notes
VOCABULARY
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  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  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 .
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.
Your Notes
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.
Checkpoint Graph y > x2  2.
1.
Example 2
Graph a system of quadratic inequalities
Graph the system of quadratic inequalities.
y > x2  2
Inequality 1
2
y  x  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.
Your Notes
Checkpoint Graph the system.
2. y < x2 + 3
y  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.
X
5
4
3
2
l
X2  3x  4
_6_
_0_
_4_
_6_
_6_
0
1
2
_4_
_0_
_6_
X
X2 + 3x  4
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_.
Checkpoint Complete the following exercise.
3. Solve the quadratic inequality x2 + 2x  3 using a table.
x
4
3
2
1
x2  2x  3
5
0
3
4
x
0
1
2
x2  2x  3
3
0
5
3  x  1
Your Notes
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
5  (  5) 2  4(  3)( 3)
x
2 (  3)
5  61
6
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  0.47_.
x
Checkpoint Complete the following exercise.
4. Solve the quadratic inequality 2x2 + 3x  5 by graphing.
x  2.5 or x  1
Your Notes
Example 5
Solve a quadratic inequality algebraically
Solve x2  x  12.
First, write and solve the equation obtained by replacing  with _=_.
_x2 + x = 12_
Write corresponding equation.
2
_x + 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  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  12_ _(0)2 + (0) = 0 ≱ 12_
Test x = _5_:
_(5)2 + (5) = 30  12_
The solution is _x  4 or x  3_.
Checkpoint Solve the inequality algebraically.
5. 2x2 + 12x  16
x  2 or x  4
Homework
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