1. Solving Linear Inequalities
2.6
Solving Linear Inequalities
1. Represent solutions to inequalities graphically and using set notation.
2. Solve linear inequalities.

2. Inequalities Inequality always points to the smaller number.
<
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
4 > 4
x > 4 is the same as {5, 6, 7…}
True
False
False
Represent inequalities: Graphically
Interval Notation
Set-builder Notation

3. 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

4. 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)

5. The set of all x such that x is greater than or equal to 5.
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
x < 2
( –∞, 2)
{ x | }
x ≥ 2
[2, ∞)
{ x | x ≥ 2}
4 < x < 9
(4, 9)
{ x | 4 < x < 9}

6. Inequalities x ≥ 5 x < –3
Graph, then write interval notation and set-builder notation.
x ≥ 5 [
Interval Notation:
[ 5, ∞)
Set-builder Notation:
{ x | x ≥ 5}
x < –3 )
Interval Notation:
(– ∞, –3)
Set-builder Notation:
{ x | x < –3 }

7. Inequalities 1 < a < 6 –7 < x ≤ 3
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 }

8. Inequalities The Addition Principle of Inequality
4 < 5
4 < 5
4 + 1 < 5 + 1
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 .

9. Inequalities
4 < 5
4 < 5
4 (–2) < 5 (–2)
4 (2) < 5 (2)
–8 < –10
–8 > –10
8 < 10
True
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 .

10. Solving Inequalities
If we multiply (or divide) by a negative, reverse the direction of the inequality!!!!!

11. Solving Inequalities
Solve then graph the solution and write it in interval notation and set-builder notation.
Don’t write = ! (
Interval Notation:
( 1, ∞ )
Set-builder Notation:
{ x | x > 1 }

12. Solving Inequalities
Solve then graph the solution and write it in interval notation and set-builder notation. ]
Interval Notation:
(– ∞, –3 ]
Set-builder Notation:
{ k | k ≤ –3 }

13. Solving Inequalities
Solve then graph the solution and write it in interval notation and set-builder notation. )
Interval Notation:
(– ∞, 6 )
Set-builder Notation:
{ p | p < 6 }

14. Solving Inequalities
Solve then graph the solution and write it in interval notation and set-builder notation.
Moving variable to the right. [
Interval Notation:
[– 3, ∞ )
Set-builder Notation:
{ m | m ≥ – 3 }