U4.7 Graph and Solve
Quadratic Inequalities
Practice 1-6: Factoring
Factor each problem completely!
1. 3 x  24 x  45
1. 3( x  3)( x  5)
2. - 2 x  128
2.  2( x  4)( x  4 x  16)
3. 3 a  27 b
3. 3( a  3 b )( a  3 b )
2
3
2
2
2
4. 64 x  27
4. (4 x  3)(16 x  12 x  9)
5. a x  ax
5. ax (1  x )(1  x  x )
2
3
2
4
6. 2 x  4 x  2 x
4
3
2
6. 2 x ( x  1)
2
2
Two Variable Quadratic Inequalities
Two Variable Quadratic Inequalities
y  ax  bx  c
2
y  ax  bx  c
2
y  ax  bx  c
2
y  ax  bx  c
2
Note: All of these are examples of two
variable quadratic inequalities written
in standard form. Quadratic
inequalities might be expressed in
factored or vertex form as well.
Steps for Graphing Two Variable
Quadratic Inequalities
To graph a two variable quadratic inequality:
1. Graph the quadratic equation. Make a dashed line
for < or >. Make solid for < or >.
2. Test a point (x, y) inside the parabola to determine
whether or not the point is a solution of the
inequality.
3. Shade the region inside the parabola if the point
from step 2 is a solution. Shade outside the
parabola if the point from step 2 is not a solution.
Remember Quadratic graphs must have 5 points
when graphing and must include the vertex!
Example 1
Graph:
y  x  3x  4
2
Test: Pick a point : (0,0)
y  x  3x  4
2
0  0  3(0)  4
2
0  4
T rue , solution
Shade the region where the true solution
will be. (inside the parabola)
Example 2: Real Life
• A manila rope used for rappelling down a cliff can
safely support a weight W (in pounds) provided:
W  14 8 0d
where d is the rope’s diameter (in
inches). Graph the inequality.
2
Graph will only be in 1st Quadrant
because the diameter of the rope can
not be negative. Less than or equal to
means the curve will be solid and
shaded below.
Be sure to test a point to check that
the shading is in the proper place.
Example 3: System of Quad. Ineq.
• Graph the system of quadratic inequalities.
y  x
y x
2
2
4
 2x  3
1.
Graph the red inequality
2.
Graph the black inequality
3.
Identify the region where the
two graphs overlap. This
region is the graph of the
system.
4.
Are the boundary lines part of
the solution? Only the red
ones!
Graph the inequality
Practice. Graph the inequality.
1. y  x 2  2 x  8
2. y  2 x 2  3 x  1
One variable Quadratic Inequalities
Different forms:
ax  bx  c  0
2
ax  bx  c  0
2
ax  bx  c  0
2
ax  bx  c  0
2
Note: All of these are examples of one
variable quadratic inequalities written
in standard form. Quadratic
inequalities might be expressed in
factored or vertex form as well.
Steps for Solving One Variable
Quadratic Inequalities
To solve a one variable quadratic inequality:
1.
2.
3.
4.
Move all terms to one side.
Try to factor, or use the quadratic formula.
Test points on each side of the critical x –values.
Determine where the values satisfy the inequality.
Remember Quadratic graphs must have 5 points
when graphing and must include the vertex!
Example 4: Solve Algebraically
x
x
2
2
 2 x  15
 2 x  15  0
( x  5)( x  3)  0
X  5,  3
-3
-3
Move all terms to one side.
2.
Try to factor, or use the
quadratic formula
3.
Test points on each side of
the critical x –values.
4.
Determine where the values
satisfy the inequality.
5
+
-
+
1.
5
The solution is:
x < -3 or x > 5.
Example 5: Solve Using a Table
x
2
x  6
x 6  0
1. Move everything to one side.
2. Make a table by hand or by using
your calculator.
3. Use the table to find when the
function will be less than or equal to 0.
< 0 or = 0
x
2
The solution of the inequality will be the x
values that make the function  0.
So,
The solution of the inequality is :
3  x  2