Download Solving & Graphing Quadratic Inequalities

Aim: How do we solve quadratic
inequalities?
Do Now:
What are the roots for
y = x2 - 2x - 3?
Aim: Quadratic Inequalities
Course: Adv. Alg. & Trig.
Graph y = x2 - 2x - 3
Finding the roots/zeroes:
Graphically:
where
the parabola
crosses
the x-axis
Algebraically:
x-intercepts
-1,0
y = 0 represents the x-axis
and the solution to
quadratic x2 - 2x - 3 = 0 is
found at the intersection of
the parabola
x-axis
Aim:and
Quadratic
Inequalities
3,0
factor
and solve
for x.
x-axis
0 = x2 - 2x - 3
0 = (x - 3)(x + 1)
x = 3 and x = -1
Course: Adv. Alg. & Trig.
Graphing a Linear Inequality
Graph the inequality y - 2x > 2
1. Convert to standard form
2. Create Table of Values
x 2 + 2x
y
0
2 + 2(0)
2
1
2 + 2(1) 4
2
2 + 2(2)
y - 2x > 2
+2x
+2x
y
> 2 + 2x
(-2,4)
Note: the
line is now
solid.
6
3. Shade the region
above the line.
4. Check the solution by
choosing a point in the
shaded region to see if it
satisfies the inequality
Aim: Quadratic Inequalities
y - 2x > 2
4 - 2(-2) > 2
4 - (-4) > 2
8>2
Course: Adv. Alg. & Trig.
Graphing a Linear Inequality
An inequality may contain
one of these four symbols:
<, >, >, or <.
y < mx + b
y > mx + b
The
The
boundary
boundary
line
line
is is
not
part
part
ofof
the
the
solution. It is drawn as a
DASHED
SOLID line.
line.
Aim: Quadratic Inequalities
Course: Adv. Alg. & Trig.
Aim: How do we solve quadratic
inequalities?
Do Now:
Graph: y – 3x < 3
Solve: y – 3x2 = 9x – 12
Aim: Quadratic Inequalities
Course: Adv. Alg. & Trig.
Quadratic Inequalities - Graphically
y > x2 - 2x - 3 & y < x2 - 2x - 3
y > x2 - 2x - 3
y < x2 - 2x - 3
(x, y)
(x, y)
-1 < x < 3
-1,0
3,0
> Shaded inside the curve
x < -1 or x > 3
-1,0
3,0
< Shaded outside the curve
What values of x satisfy these inequalities when y = 0
0 > x2 - 2x - 3
0 < x2 - 2x - 3
Aim: Quadratic Inequalities
Course: Adv. Alg. & Trig.
The values of x found within the shaded regions.
Graph y ≥ x2 - 1 or x2 - 1 ≤ y
Graphically:
Because y is
greater than or
equal to (≥) x2 - 1,
the parabola is
shaded inside the
curve and
includes the curve
itself
-1,0
1,0
x-axis
roots
What values of x satisfy the
quadratic inequality when y = 0?
-1 ≤ x ≤ 1
Aim: Quadratic Inequalities
(x2 – 1 = 0)
Course: Adv. Alg. & Trig.
Exceptions - 1
0y > x2 - 4x + 4
y > x2 - 4x + 4
Quadratic Inequalities that
have roots that are equal
0y=> x2 - 4x + 4 = (x - 2)(x - 2)
x = 2 root/zero
What values of x satisfy the
quadratic inequality
0 > x2 - 4x + 4?
(2,0)
0y < x2 - 4x + 4
Solution: {x| x = 2}
What values of x satisfy the
quadratic inequality
0 < x2 - 4x + 4?
Solution: 
Aim: Quadratic Inequalities
(2,0)
Course: Adv. Alg. & Trig.
Exceptions - 2
y > x2 + 1
Quadratic Inequalities
that have no roots.
What values of x satisfy the
quadratic inequality
0 > x2 + 1?
(0, 1)
Solution: {x| x = }
y < x2 + 1
What values of x satisfy the
quadratic inequality
0 < x2 + 1?
(0, 1)
Solution: {x| x = }
Aim: Quadratic Inequalities
Course: Adv. Alg. & Trig.
General Solutions
of Quadratic Inequalities where
a > 0 and r1 < r2
(r1 and r2 are the unequal roots)
Quadratic
Inequality
ax2 + bx + c < 0
Solution
Interval
r1 < x < r2
ax2 + bx + c < 0
r 1 < x < r2
ax2 + bx + c > 0
r1 < x or
x > r2
r1 < x or
x > r2
ax2 + bx + c > 0
Aim: Quadratic Inequalities
Graph of
Solution
r1
r2
r1
r2
r1
r2
r1
r2
Course: Adv. Alg. & Trig.
Critical Numbers & Test Intervals
x2 – 2x – 3 < 0
f x = x2 -2 x-3
44
(x + 1)(x – 3) = 0
x = -1 and x = 3
are the roots or
the zeros that
create 3 test
intervals
roots, or
zeros
22
-5
-5
(-, -1)
(-1,0)
(-1, 3)
(3, )
(3,0)
55
-2
-2
Critical Numbers for
testing the inequality
-4
-4
Test
Interval
Representative
x-value
(-, -1)
(-1, 3)
(3,  )
x = -3
x=0
x=5
Aim: Quadratic Inequalities
Value of Polynomial
Is this value < 0?
(-3)2 – 2(-3) – 3 = 12 No
(0)2 – 2(0) – 3 = -3 Yes
(5)2 – 2(5) – 3 = 12 No
Course: Adv. Alg. & Trig.
Model Problems
Solve
algebraically and
Graph:
y < x2 – 12x + 27
0 < x2 – 12x + 27
Test
Interval
Representative
x-value
Aim: Quadratic Inequalities
Value of Polynomial
Course: Adv. Alg. & Trig.
Model Problems
Graph the
solution set for
x2 – 2 > -x – 3
Aim: Quadratic Inequalities
Course: Adv. Alg. & Trig.
Regents Question
Which graph best represents the inequality
y + 6 > x2 – x?
1)
6
6
4
4
2
2
-5
5
2)
-5
-2
5
-2
-4
-4
-6
-6
-8
-8
6
6
4
4
2
2
3)
-5
5
-2
-4
-6
Aim: Quadratic Inequalities
-8
4)
-5
5
-2
-4
-6
Course: Adv. Alg. & Trig.
-8
Model Problems
Graph the
solution set for
2(x – 2)(x + 3) <
(x – 2)(x + 3)
Aim: Quadratic Inequalities
Course: Adv. Alg. & Trig.
Solve 0 ≥ x2 - 1 algebraically
Algebraically:
0 ≥ x2 - 1
0 ≥ (x - 1)(x + 1)
0 ≥ (x - 1) 0 ≥ (x + 1)
x ≥ 1 and x ≥ -1
?
-1,0
Because 0 ≥ x2 - 1,
1,0 2
• x - 1 must be a negative number or 0
roots
• a negative number is the product of
a positive & negative #
-1 ≤ x ≤ 1
one of the factors must be positive
and the other negative
Aim: Quadratic
Alg. & Trig.
If ab < 0, then
a < Inequalities
0 and b > 0, orCourse:
a > Adv.
0 and
b < 0.
Solve 0 ≥ x2 - 1 algebraically (con’t)
Set quadratic = 0 x2 - 1 ≤ 0
(x - 1)(x + 1) ≤ 0
Factor
(x - 1) ≥ 0 and (x + 1) ≤ 0
EXTRANEOUS
x ≥ 1 and x ≤ -1
(x - 1) ≤ 0 and (x + 1) ≥ 0
x ≤ 1 and x ≥ -1
KEY WORD - “and” 
What values of x can satisfy both inequalities
for each set?
x CANNOT be a
number less than or
equal to -1 and greater
than or equal to 1.
“What values of x are
less than or equal to 1
and greater than or
equal to -1?”
-1 ≤ x ≤ 1
Aim: Quadratic Inequalities
Course: Adv. Alg. & Trig.
Aim: How do we solve quadratic
inequalities?
Do Now:
Graph the inequality
y ≥ x2 – 1
Aim: Quadratic Inequalities
Course: Adv. Alg. & Trig.