1. U1A L6 Linear, Quadratic & Polynomial Inequalities
UNIT 1A LESSON 6
Linear, Quadratic and Polynomial
Inequalities

2. INTERVALS & INEQUALITIES
U1A L6 Linear, Quadratic & Polynomial Inequalities
INTERVALS & INEQUALITIES
Interval
Notation
Inequality
Notation
Graph

3. U1A L6 Linear, Quadratic & Polynomial Inequalities
Express the following intervals in terms of inequalities
and graph the intervals

4. U1A L6 Linear, Quadratic & Polynomial Inequalities
REMEMBER:
Linear inequalities are solved the same as equations
EXCEPT
when the final step involves dividing by a NEGATIVE.
You must change the direction of the sign.

5. U1A L6 Linear, Quadratic & Polynomial Inequalities
Linear Inequalities
Solve the inequalities.
State the answer in inequality
form, interval form and graph.
EXAMPLE 2
5x + 7 > – 8
5x > – 15
x > – 3 inequality form
[– 3, ∞) interval form
(−𝟑, −𝟖)
𝒀 𝟐 =−𝟖
𝒀 𝟐 =𝟓𝒙 +𝟕 -3
graph

6. U1A L6 Linear, Quadratic & Polynomial Inequalities
Linear Inequalities
Solve the inequalities.
State the answer in inequality form, interval form and graph.
EXAMPLE 3
𝒀 𝟏 =𝟑𝒙 +𝟏
3x + 1 < 7x – 7
– 4x < – 8
x > 2 inequality form
(𝟐, 𝟕)
(2, ∞) interval form 2
graph
𝒀 𝟐 =𝟕𝒙−𝟕

7. U1A L6 Linear, Quadratic & Polynomial Inequalities
Linear Inequalities
Solve the inequalities and graph.
State the answer in inequality form and interval form.
𝒀 𝟏 =𝟖−𝒙
𝒀 𝟐 =𝟓𝒙+𝟐
EXAMPLE 4
8 – x > 5x + 2
(𝟏, 𝟕)
– 6x > – 6
−𝟔𝒙 −𝟔 < −𝟔 −𝟔
x < 1 inequality form
(–∞, 1) interval form 1
graph

8. SOLVING POYNOMIAL INEQUALITIES
U1A L6 Linear, Quadratic & Polynomial Inequalities
SOLVING POYNOMIAL INEQUALITIES
In order to solve any polynomial equation or inequality you must FACTOR first!!!
EXAMPLE 5 : x2 + 2x – 8 = 0
Let’s use
our heads
(x + 4)(x – 2) = 0
x = –4 or x = 2
If x < – 4
If x is between –4 and 2
If x > 2
( )( – 2) is
( )( – 2) is
( )( – 2) is
−𝟏𝟎 −𝟓
−𝟏𝟎 𝟎 𝟎 𝟔 𝟑 𝟑 𝟔 −𝟓
negative
positive
positive + − + 2 –4

9. SOLVING POYNOMIAL INEQUALITIES
U1A L6 Linear, Quadratic & Polynomial Inequalities
SOLVING POYNOMIAL INEQUALITIES
x2 + 2x – 8 = 0
x2 + 2x – 8 > 0
x2 + 2x – 8 < 0
(x + 4)(x – 2) = 0
(x + 4)(x – 2) > 0
(x + 4)(x – 2) < 0
x = – 4 or x = 2
x < – 4 or x > 2
– 4 < x < 2
−∞,−4 ∪ 2, ∞
(−𝟒, 𝟐) + + − 2
– 4

10. 𝒀 𝟏 =𝒙𝟐 + 𝟐𝒙 – 𝟖
𝒙<−𝟒
𝒙>𝟐
𝒙=−𝟒
𝒙=𝟐
−𝟒<𝒙<𝟐

11. U1A L6 Linear, Quadratic & Polynomial Inequalities
EXAMPLE 6 Factor x3 + 2x2 – 5x – 6
Test potential zeros ±1, ±2, ±3, ±6.
23 + 2(2)2 – 5(2) – 6 = – 10 – 6 = 0
(x – 2) is a factor
Subtraction Method
Addition Method
x3 + 2x2 – 5x – 6 = (x – 2)(1x2 + 4x + 3)
= (x – 2)(x + 3)(x + 1)

12. U1A L6 Linear, Quadratic & Polynomial Inequalities
EXAMPLE 6 continued x3 + 2x2 – 5x – 6
x3 + 2x2 – 5x – 6
= (x – 2)(x + 3)(x + 1) − + + −
– 3
– 1 2
If x < – 3
( – 2)( )( ) is −𝟒
neg −𝟒 −𝟒
If – 3 < x < – 1
( – 2)( )( ) is
pos −𝟐 −𝟐 −𝟐
If – 1 < x < 2
( – 2)( )( ) is
neg 𝟎 𝟎 𝟎
If x > 2
( – 2)( )( ) is 𝟓 𝟓
pos 𝟓

13. U1A L6 Linear, Quadratic & Polynomial Inequalities
EXAMPLE x3 + 2x2 – 5x – 6
EXAMPLE 6 x3 + 2x2 – 5x – 6
x3 + 2x2 – 5x – 6
= (x – 2)(x + 3)(x + 1)
neg
pos
pos
neg –3 –1 2
x3 + 2x2 – 5x – 6 < 0
x3 + 2x2 – 5x – 6 > 0
x < –3 or –1 < x < 2
–3 < x < –1 or x > 2
−∞, −𝟑 ∪ −𝟏, 𝟐
−𝟑, 𝟏 ∪[𝟐, ∞)

14. x > 2 –3 < x < –1 –1 < x < 2 x < –3
𝒙 𝟑 +𝟐 𝒙 𝟐 −𝟓𝒙−𝟔≥𝟎
x > 2
–3 < x < –1
𝒙=−𝟑
𝒙=−𝟏
𝒙=𝟐
–1 < x < 2
x < –3
𝒙 𝟑 +𝟐 𝒙 𝟐 −𝟓𝒙−𝟔≤𝟎