MATH 1100

SECTION 3.6 Notes

Linear Inequalities – Text Pages 181-185
RULES FOR INEQUALITIES
Rule
/
Description
1. / Adding the same quantity to each side of an inequality gives an equivalent inequality.
2. / Subtracting the same quantity to each side of an inequality gives an equivalent inequality.
3. If , then / Multiplying each side of an inequality by the same positive quantity gives an equivalent inequality.
4. If , then / Multiplying each side of an inequality by the same negative quantity reverses the direction of the inequality.
5. If and , then / Taking reciprocals of each side of an inequality involving positive quantities reverses the direction of the inequality.
6. If and , then / Inequalities can be added.

Interval Notation:

Interval notation uses parenthesis and brackets.

Open Interval, (a, b): If a and b are real numbers such that a < b, we define an open interval (a, b) as the set of all numbers x for which , so

.

(NOTE: Be CAREFUL not to confuse the ordered pair notation with the open interval notation!!)

Closed Interval, [a, b]: If a and b are real numbers such that a £ b, we define an closed interval [a, b] as the set of all numbers x for which , so

.

Half-Open Interval, [a, b) or (a, b]:

So, the solution to the inequality x < 2 can be written as .

NOTE: We never include infinity in a solution (i.e., always use parenthesis around infinity!)


Example 1: Solve and graph the following inequality:

Interval:

Graph:

Example 2: Solve and graph the following inequality:

Interval:

Graph:


Example 3: Solve and graph the following inequality:

Interval:

Graph:

Example 4: Solve and graph the following inequality:

Interval:

Graph:


Example 5: A telephone company offers two long-distance plans:

Plan A: $25 per month and $0.05 per minute

Plan B: $5 per month and $0.12 per minute

For how many minutes of long-distance calls would plan B be financially advantageous?

Identify the variable:

Express Unknowns:

Set Up Model:

Solve:

Check:

State: