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Aim: Quadratic Inequalities Course: Adv. Alg. & Trig.
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
wherethe parabola
crossesthe x-axis
Finding the roots/zeroes:
Graphically:
-1,0 3,0x-axis
y = 0 represents the x-axis and the solution to
quadratic x2 - 2x - 3 = 0 is found at the intersection of
the parabola and x-axis
0 = x2 - 2x - 30 = (x - 3)(x + 1)x = 3 and x = -1
factorand solve
for x.
Algebraically:
x-intercepts
Aim: Quadratic Inequalities Course: Adv. Alg. & Trig.
Graphing a Linear Inequality
Graph the inequality y - 2x > 2 y - 2x > 21. Convert to standard form +2x +2x
y > 2 + 2x2. Create Table of Values
y =
2 +
2x
3. Shade the region above the line.
x y2 + 2x
0
1
2
2
4
6
2 + 2(0)
2 + 2(1)
2 + 2(2)
4. Check the solution by choosing a point in the shaded region to see if it satisfies the inequality
(-2,4)
4 - 2(-2) > 24 - (-4) > 2 8 > 2
y - 2x > 2
Note: the line is now
solid.
Aim: Quadratic Inequalities 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 boundary line is part of the solution. It is drawn as a
SOLID line.
The boundary line is not part of the solution. It is drawn as a
DASHED line.
y < mx + b y > mx + b
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.
y > x2 - 2x - 3 y < x2 - 2x - 3
Quadratic Inequalities - Graphically y > x2 - 2x - 3 & y < x2 - 2x -
3
-1,0 3,0 -1,0 3,0
-1 < x < 3 x < -1 or x > 3
> Shaded inside the curve < Shaded outside the curve
The values of x found within the shaded regions.0 > x2 - 2x - 3 0 < x2 - 2x - 3
What values of x satisfy these inequalities when y = 0
(x, y)
(x, y)
Aim: Quadratic Inequalities Course: Adv. Alg. & Trig.
Graph y ≥ x2 - 1 or x2 - 1 ≤ y
Because y is greater than or
equal to (≥) x2 - 1, the parabola is
shaded inside the curve and
includes the curve itself
Graphically:
-1 ≤ x ≤ 1
What values of x satisfy the quadratic inequality when y = 0?
x-axis
(x2 – 1 = 0)
-1,0 1,0
roots
Aim: Quadratic Inequalities Course: Adv. Alg. & Trig.
Exceptions - 1
What values of x satisfy the quadratic inequality
0 > x2 - 4x + 4?
y < x2 - 4x + 4 Solution: {x| x = 2}
y > x2 - 4x + 4x = 2 root/zero
What values of x satisfy the quadratic inequality
0 < x2 - 4x + 4?
y > x2 - 4x + 4
Solution:
= (x - 2)(x - 2)0 =
Quadratic Inequalities that have roots that are equal
(2,0)
(2,0)
0
y > x2 - 4x + 4
0
Aim: Quadratic Inequalities Course: Adv. Alg. & Trig.
Exceptions - 2
Quadratic Inequalities that have no roots.
What values of x satisfy the quadratic inequality
0 > x2 + 1?
Solution: {x| x = }
What values of x satisfy the quadratic inequality
0 < x2 + 1?
y > x2 + 1
Solution: {x| x = }(0, 1)
y < x2 + 1
(0, 1)
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
Solution Interval
Graph of Solution
ax2 + bx + c < 0 r1 < x < r2
ax2 + bx + c < 0 r1 < x < r2
ax2 + bx + c > 0 r1 < x or
x > r2
ax2 + bx + c > 0 r1 < x or
x > r2
r1 r2
r1 r2
r1 r2
r1 r2
Aim: Quadratic Inequalities Course: Adv. Alg. & Trig.
4
2
-2
-4
-5 5
f x = x2-2x-3
Critical Numbers & Test Intervals
x2 – 2x – 3 < 0
(-1,0) (3,0)
roots, orzeros
Critical Numbers fortesting the inequality
(x + 1)(x – 3) = 0
x = -1 and x = 3 are the roots or the zeros that create 3 test
intervals
(-, -1)
(-1, 3)
(3, )
Test Interval
Representative x-value
Value of Polynomial
x = -3 (-3)2 – 2(-3) – 3 = 12
x = 0 (0)2 – 2(0) – 3 = -3
(-, -1)
(-1, 3)
(3, ) x = 5 (5)2 – 2(5) – 3 = 12
4
2
-2
-4
-5 5
f x = x2-2x-3
(-1, 3)
Is this value < 0?
NoYes
No
Aim: Quadratic Inequalities Course: Adv. Alg. & Trig.
Model Problems
Test Interval
Representative x-value
Value of Polynomial
Solve algebraically and Graph:
y < x2 – 12x + 27
0 < x2 – 12x + 27
Aim: Quadratic Inequalities 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?
6
4
2
-2
-4
-6
-8
-5 5
6
4
2
-2
-4
-6
-8
-5 5
-5 5
6
4
2
-2
-4
-6
-8
-5 5
6
4
2
-2
-4
-6
-8
1) 2)
3) 4)
Aim: Quadratic Inequalities Course: Adv. Alg. & Trig.
Model Problems
Graph the solution set for
2(x – 2)(x + 3) < (x – 2)(x + 3)
Aim: Quadratic Inequalities Course: Adv. Alg. & Trig.
Because 0 ≥ x2 - 1, • x2 - 1 must be a negative number or 0
Solve 0 ≥ x2 - 1 algebraically
-1 ≤ x ≤ 1
Algebraically:
0 ≥ x2 - 1
0 ≥ (x - 1)(x + 1)
x ≥ 1 and x ≥ -1
?0 ≥ (x - 1) 0 ≥ (x + 1)
• a negative number is the product of a positive & negative #
one of the factors must be positive and the other negative
If ab < 0, then a < 0 and b > 0, or a > 0 and b < 0.
-1,0 1,0
roots
Aim: Quadratic Inequalities Course: Adv. Alg. & Trig.
Solve 0 ≥ x2 - 1 algebraically (con’t)x2 - 1 ≤ 0
(x - 1)(x + 1) ≤ 0
(x - 1) ≥ 0 (x + 1) ≤ 0 (x - 1) ≤ 0 (x + 1) ≥ 0
x ≥ 1 x ≤ -1 x ≤ 1 x ≥ -1
and
and
and
and
x CANNOT be a number less than or
equal to -1 and greater than or equal to 1.
EXTRANEOUS
Set quadratic = 0
Factor
What values of x can satisfy both inequalities for each set?
“What values of x are less than or equal to 1
and greater than or equal to -1?”
-1 ≤ x ≤ 1
KEY WORD - “and”
Aim: Quadratic Inequalities Course: Adv. Alg. & Trig.
Aim: How do we solve quadratic inequalities?
Do Now:
Graph the inequality
y ≥ x2 – 1