Solving Linear Inequalities
Chapter 2, Sections 5
- Properties Used to Solve Inequalities
- For real numbers a, b, and c:
- If a > b, then a + c > b + c.
- If a > b, then a - c > b - c.
- If a > b and c > 0, then ac > bc.
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| If a > b and c > 0, then |
a
c |
> |
b
c |
- If a > b and c < 0, then ac < bc.
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| If a > b and c < 0, then |
a
c |
< |
b
c |
- Note that when you multiply by a negative number the inequality sign changes direction.
- How to Write Solutions
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In Set Builder
Notation
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On Number Line
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In Interval
Notation
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| x > a |
{ x | x > a } |
 |
(a,  ) |
x  a |
{ x | x a } |
 |
[a, ) |
| x < a |
{ x | x < a } |
 |
(-, a) |
x  a |
{ x | x a } |
 |
(-, a] |
| a < x < b |
{ x | a < x < b } |
 |
(a, b) |
| a x b |
{ x | a x b } |
 |
[a, b] |
| a < x b |
{ x | a < x b } |
 |
(a, b] |
| a x < b |
{ x | a x < b } |
 |
[a, b) |
- Solving Simple Inequalities
- Solve a simple linear inequality like you would solve a linear equation except that the = sign is repaced by an inequality sign.
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| Example |
Solution Steps |
| Solve 5x - 2 < 3 |
5x < 5
x < 1 |
The solution is x < 1
|
- Solving Compound Inequalities
- A compound inequality has an expression in the middle sandwiched between two inequalities
like in 5 < 2x - 3 11.
- Goal: get x alone (with coefficeint of 1) in the middle of the sandwich with numbers on the outsides.
- What you do to the middle (while trying to get x alone) you must do to the outside expressens as well. That will keep each step's inequality equivalent to the original inequality (equivalent means that the solutions are the same).
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| Example: Solve 5 < 2x - 3 < 11
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Solution Steps |
In Words... |
| 8 < 2x < 14 |
Add 3 to each expression. |
| 4 < x < 7 |
Divide each expression by 2. |
The solution is 4 < x < 7
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