Transcript Section 2.6
2.6
Solving Linear Inequalities
1. Represent solutions to inequalities graphically and using set notation.
2. Solve linear inequalities.
Inequalities < > is less than is greater than is less than or equal to is greater than or equal to Inequality always points to the smaller number.
True or False?
4 4 True 4 > 4 False x > 4 is the same as {5, 6, 7…} False Represent inequalities: Graphically Interval Notation Set-builder Notation
Graphing Inequalities If the variable is on the left, the arrow points the
same direction
as the inequality.
Parentheses/bracket method : Parentheses: endpoint is not included <, > Bracket: endpoint is included ≤, ≥ x < 2 x ≥ 2 Open Circle/closed circle method: Open Circle: endpoint is not included <, > Closed Circle: endpoint is included ≤, ≥ x < 2 x ≥ 2
Inequalities – Interval Notation [( smallest, largest )] Parentheses: endpoint is not included <, > Bracket: endpoint is included ≤, ≥ Infinity:
always
uses a parenthesis x < 2 ( –∞, 2) x ≥ 2 [2, ∞) 4 < x < 9 3-part inequality (4, 9)
Inequalities – Set-builder Notation {variable | condition } pipe {
x
|
x
5 } The set of all
x
such that
x
is greater than or equal to 5.
x < 2 ( –∞, 2) < 2 x ≥ 2 [2, ∞) { x | x ≥ 2} 4 < x < 9 (4, 9) { x | 4 < x < 9}
Inequalities Graph, then write interval notation and set-builder notation.
x ≥ 5
[
Interval Notation: Set-builder Notation: [ 5, ∞) { x | x ≥ 5} x < –3
)
Interval Notation: (– ∞, –3) Set-builder Notation: { x | x < –3 }
Inequalities Graph, then write interval notation and set-builder notation.
1 < a < 6
( )
Interval Notation: ( 1, 6 ) Set-builder Notation: { a | 1 < a < 6 } –7 < x ≤ 3
( ]
Interval Notation: (– 7, –3] Set-builder Notation: { x | –7 < x ≤ 3 }
Inequalities 4 < 5 4 + 1 < 5 + 1 4 < 5 4 – 1 < 5 – 1 5 < 6 3 < 4 True True
The Addition Principle of Inequality
If
a
<
b
, then
a +
c
<
b
+
c
for all real numbers
a, b,
and
c
. Also true for >, , or .
Inequalities 4 < 5 4 (2) < 5 (2) 8 < 10 True 4 < 5 4 (–2) < 5 (–2) –8 < –10 –8 > –10 False If we multiply (or divide) by a negative, reverse the direction of the inequality!!!!!
The Multiplication Principle of Inequality
If
a < b,
then
ac
<
bc
if
c
is a positive real number.
If
a
<
b
, then
ac
>
bc
if
c
is a negative real number . The principle also holds true for >, , and .
Solving Inequalities If we multiply (or divide) by a negative, reverse the direction of the inequality!!!!!
4
x
4 16 4
x
4 4 4
x
16 4
x
4 4
x
4 16 4
x
4
Solving Inequalities
Solve
then
graph
the solution and write it in
interval notation
and
set-builder notation
.
3 4 Don’t write = !
3
x x
4 3 7 4 3 3
x
1
(
Interval Notation: ( 1, ∞ ) Set-builder Notation: { x | x > 1 }
Solving Inequalities
Solve
then
graph
the solution and write it in
interval notation
and
set-builder notation
.
4 9
k
4
k
4 4
k k
19 4 4 5
k
19 4 5
k
5 15 5
k
3
]
Interval Notation: Set-builder Notation: (– ∞, –3 ] { k | k ≤ –3 }
Solving Inequalities
Solve
then
graph
the solution and write it in
interval notation
and
set-builder notation
.
5 3
p
10 3 3 5
p
3 10 5
p
5
p
6 30 5
)
Interval Notation: (– ∞, 6 ) Set-builder Notation: { p | p < 6 }
Solving Inequalities
Solve
then
graph
the solution and write it in
interval notation
and
set-builder notation
.
1 5 6
m
7 1 2 3
m
1 10 2 1 5 6
m
6
m
7 7 5 3
m
10 1 2 1 3
m
1 12
m
15
m
14 15
m
15
m
5 3
m
14 14 5 14
Moving variable to the right.
12
m
14 15
m
5 12
m
12
m
14 5 3
m
5 5 9 3 3
m
3
m
3
[
3
m
3 9 3 Interval Notation: [– 3, ∞ ) Set-builder Notation: { m | m ≥ – 3 }
m
3