Inequality Lecture Notes
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To solve simple linear inequalities, we isolate the variable as in equations:
Notation Continued:
| Interval Notation: | Graph: | |
| x < a | (-∞, a) | |
| x < a | (-∞, a] | |
| a < x | (a, ∞) | |
| x > a | [a, ∞) | |
| a < x < b | [a, b) | |
| a < x < b | (a, b] | |
Two more notational notes:
1.) {x| a < x < b} is read as :
"The set of all x such that x is greater than or equal to a and
x is less than or equal to b." or "The set of all x such that x
is between a and b inclusive."
2.) These expressions are nonsense:
Multiplying or Dividing through by a negative:
If x > a, then -x <-a. In this illustration a is greater than zero, but it would work if a <0 also.
Anytime you multiply or divide both sides of an inequality by a negative you need to flip the inequality.
Problems involving "and" or "or":
| Examples: | graph: | simplified inequality: |
| x < 5 and x > 3 | graph | inequality |
| x < 5 and x < 3 | graph | inequality |
| x < 3 and x > 5 | graph | inequality |
| x < 5 or x > 3 | graph | inequality |
| x < 5 or x < 3 | graph | inequality |
| x < 3 or x > 5 | graph | inequality |
Problems Involving Absolute Values:
Consider the following:
| |x| > 1 ⇒ | The following is read, "For all a > 0, the absolute value of x greater than a implies"
" a > 0, |x| > a ⇒
|
| |x| ≤ 1 ⇒ | " a > 0, |x| < a
⇒
|
Solve: |x| > -2
