8.2 Linear Inequalities

We will remind ourselves how to solve inequalities and graph on a number line as well as the coordinate plane

Inequality

a < x < b

a ≤ x ≤ b

a ≤ x < b

a < x ≤ b

x > a

x ≥ a

x < a

x ≤ a

Interval

(a, b)

[a, b]

[a, b)

(a, b]

(a, ∞)

[a, ∞)

(–∞, a)

(–∞, a]

Graph

a b

a b

a b

a b

a

a

a

a

In a compound inequality, two conditions are given- conjunction- disjunction

“and”

6 – 10x < 5 – 10x < –1

3 5 5

2 4

4x

b)

the intersection of the sets “or” the union of the sets

Ex 1) Solve and graph on a number line. Express in interval notation.a)

1

10x

1,

10

110

3 ≤ 4x – 1 < 73 ≤ 4x – 14 ≤ 4x1 ≤ xx ≥ 1

*deal with one part at a time*

4x – 1 < 7 4x < 8 x < 2 [1, 2)

1 2

Ex 1) Solve and graph on a number line. Express in interval notation.c)

Absolute Value: *Remember could mean x = 2 or x = –2*Hint: ││< #

5

2x

say “less thAND”

2x – 3 ≥ 2 or 2x – 3 < –4 2x ≥ 5 2x < –1

1

2x

52

12

512 2( , ) [ , )

2x say “greatOR”││> #

Ex 2) Solve and graph on a number line. Express in interval notation.a) │6 – 4x│ ≤ 2 “and”

6 – 4x ≤ 2 –4x ≤ –4 x ≥ 1

and 6 – 4x ≥ –2 –4x ≥ –8 x ≤ 2 [1, 2]

1 2

Ex 2) Solve and graph on a number line. Express in interval notation.b) │2x + 1│ > 3 “or”

2x + 1 > 3 2x > 2 x > 1

or 2x + 1 < –3 2x < –4 x < –2

1–2

(–∞, –2) (1, ∞)

Graphing linear inequalities in two variables in the coordinate plane

< or >dotted line

≤ or ≥solid line

y < or y ≤shade “below”

y > or y ≥shade “above”

The two regions the coordinate plane is divided into is called half-plane.The line is the boundary.closed half-plane: solid line

open half-plane: dotted line

Ex 3) Graph in the coordinate planea) 3x – y > 4

–y > –3x + 4 y < 3x – 4 shade below

b) –3 ≤ 2x + y < 6

–3 ≤ 2x + y –y ≤ 2x + 3 y ≥ –2x – 3 shade above

do each line

2x + y < 6 y < –2x + 6 shade below

c) y < –│x + 2│

abs valuereflect over x-axisshifted 2 leftdotted lineshade below

Homework

#802 Pg 398 #1, 5, 9, 13, 16, 21, 24, 29, 34, 38, 39, 43, 44