1. Linear Inequalities (simple)
Corresponds to chapter 2.4 MCR

2. Linear inequality (can be nonlinear in one variable)
linear inequality: a linear relationship between a variable, a number and an inequality symbol (>,<, ≥, ≤) .

3. The “greater than” symbols
> means “greater than” --which means the solutions will be to the right of our value on the number line
≥ means “greater than or equal to” --which means the solutions will be to the right of our value on the number line but may also include our number of interest

4. The “less than” symbols
< means “less than” --which means the solutions will be to the left of our value on the number line
≤ means “less than or equal to” --which means the solutions will be to the left of our value on the number line but may also include our number of interest

5. How can we express this using set notation and the number line?
Using “x” as our variable, write the solution set in set notation for the graph.
𝑥 𝑥 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟>−1
The set contains the variable x such that x includes all real numbers greater than -1

6. One more
Using “x” as our variable, write the solution set in set notation for the graph.
{𝑥|𝑥≤0}
“The set x such that x includes all real numbers that are less than or equal to zero”

7. Interval notation We can express the same ideas with interval notation
Interval notation uses brackets “[ ] “ to designate equal to
Interval notation uses parenthesis “() “ to indicate the value is not equal to.
The symbol −∞, +∞ designate negative and positive infinity.

8. Interval notation on the number line
Which means [𝟒,∞)
“the solution includes all real numbers greater than or equal to 4”

9. Another example
Which means (−∞,−2) or the solutions include all real numbers less than -2 (but not including -2) all the way to negative infinity (which is never reached).

10. A few example problems −4>3𝑛+5 (subtract 5 from both sides)
−9>3𝑛 (divide both sides by 3)
−3>𝑛
Writing the solution in interval notation (−∞,−3)
In set notation {𝑛|𝑛<−3}

11. Another (I d0) −5+6𝑥≥7 (add +5 to both sides)
6𝑥≥12 (divide both sides by 2)
𝑥≥2
[2,∞) (in interval notation)
{𝑥|𝑥≥2} (in set notation)

12. A caveat
If we multiply or divide the inequality by a negative number, we must change the direction of the inequality symbol!

13. Try one −10≥−3𝑥+ 2 (subtract 2 from both sides)
−12≥−3𝑥 ( divide both sides by -3x and change the direction of the inequality operator).
4≤𝑥
[4, ∞) (interval notation)
{𝑥|𝑥≥4} (set notation)