Download Math 95 Notes Section 2.8 Linear Inequalities Students will be able

Math 95 Notes
Section 2.8
Linear Inequalities
Students will be able to
• Solve linear inequalities
• Represent the solution set on a number line, in interval notation, and in set
builder notation
Example:
Solve: p − 4 ≤ 9 and graph the solution set. Also, write the solution set in interval and
set-builder notation.
To solve this problem, we will use that same strategies as we did with solve equations.
One thing to help with the solution, you will always want to get the variable on the left.
This will make it easier represent the solution set.
p−4≤9
p−4+4 ≤9+4
p ≤ 13
]
Number Line:
13
Interval Notation: ( −∞,13]
Set-Builder Notation: { p p ≤ 13}
Example:
Solve: −2 w < 14 and graph the solution set. Also, write the solution set in interval and
set-builder notation.
One item that we have to remember when solving inequalities, whenever you multiply or
divide by a negative, you must change the direction of the sign.
−2 w < 14
−2 w 14
>
−2
−2
w > −7
Number Line:
(
-7
Example:
1
( 4 x − 8) and graph the solution set. Also, write the solution set in
2
interval and set-builder notation.
Solve: 3 ( x + 1) − 2 ≤
1
( 4 x − 8)
2
3x + 3 − 2 ≤ 2 x − 4
3x + 1 ≤ 2 x − 4
3 ( x + 1) − 2 ≤
3x − 2 x + 1 ≤ 2 x − 2 x − 4
x + 1 ≤ −4
x + 1 − 1 ≤ −4 − 1
x ≤ −5
]
Number Line:
-5
Interval Notation: ( −∞, −5]
Set-builder Notation: { x x ≤ −5}
Example:
Solve: 3 ( 2 x + 5 ) + 2 < 5 ( 2 x + 2 ) + 3 and graph the solution set. Also, write the solution set
in interval and set-builder notation.
3 ( 2 x + 5) + 2 < 5 ( 2 x + 2 ) + 3
6 x + 15 + 2 < 10 x + 10 + 3
6 x + 17 < 10 x + 13
6 x − 10 x + 17 < 10 x − 10 x + 13
−4 x + 17 < 13
−4 x + 17 − 17 < 13 − 17
−4 x < −4
−4 x −4
>
−4 −4
x >1
Number Line:
(
1
Interval Notation: (1, ∞ )
{
Set-builder Notation: x x > 1}
Example:
1
1 5
1
Solve: w − ≤ w + and graph the solution set. Also, write the solution set in interval
3
2 6
2
and set-builder notation.
1 5
1
1
 3 w − 2 ≤ 6 w + 2  6
2 w − 3 ≤ 5w + 3
2 w − 5w − 3 ≤ 5w − 5w + 3
−3w − 3 ≤ 3
−3w − 3 + 3 ≤ 3 + 3
−3w ≤ 6
−3w 6
≥
−3 −3
w ≥ −2
[
Number Line:
Interval Notation: [ −2, ∞ )
{
-2
Set-builder Notation: w w ≥ −2}
Example:
1
x − 1 < 5 and graph the solution set. Also, write the solution in interval and
4
set-builder notation.
Solve: −3 ≤
To solve this problem, you want to work on getting the variable by itself in the middle.
1
x −1 < 5
4
1
−3 + 1 ≤ x − 1 + 1 < 5 + 1
4
1
−2 ≤ x < 6
4
1
−2 ( 4 ) ≤ x ( 4 ) < 6 ( 4 )
4
−8 ≤ x < 24
−3 ≤
Number line:
[
)
-8
24
Interval Notation: [ −8, 24 )
{
Set-builder Notation: x −8 ≤ x < 24}
Example:
Solve: −24 < −2 x < −20 and graph the solution set. Also, write the solution in interval and
set-builder notation.
−24 < −2 x < −20
−24 −2 x −20
>
>
−2
−2
−2
12 > x > 10
10 < x < 12
In this problem, we noticed that in the 4th line that the numbers are not in correct order.
So we put them in the correct order and flip the sign to match it.
Number line:
(
)
10
12
Interval Notation: (10,12 )
{
Set-builder Notation: x 10 < x < 12}