1. Pre- Calculus
Lesson 1.1 begin
Chapter 1: Functions

2. 1.1 Introduction (p. 2-3)
Copy definitions: (copy symbols if any apply)
Natural Numbers
Integers
Rational Numbers
Irrational Numbers
Real Numbers
Being closed under operation(s)
If two numbers of a particular set are under an operation, and the
result of that operation is a number still in that same set, then it is
closed under that operation.

3. Ex: 2 and 3 are natural numbers. Test all operations to see
which operations natural numbers are closed under.
2 + 3 = 5 (natural number)
3 + 2 = 5 (natural number)
Closed under addition
Note: You must check the numbers in any order.
Natural numbers are closed under: Addition and multiplication.
Integers are closed under: Addition, subtraction and multiplication.
Rational numbers are closed under: Addition, subtraction,
multiplication, and division
except for 0.

4. All real numbers can be shown on a coordinate line.
1.2 The Real Line (p. 3-12)
All real numbers can be shown on a coordinate line. -3 3
Inequalities:
The relative position of two points on a number
line is used to define an inequality.
Lesson 1.2 begins a b
We say: a is less than b or b is greater than a.
a < b
b > a

5. Inequality properties:
1. Precisely one of a < b, b < a, or a = b holds.
2. If a > b, then a + c > b + c.
ex: let a = 4, b = 1, and c = 10. Check the property.
3. If a > b and c > 0, then ac > bc.
Use the same values in the example above.
4. If a > b and c < 0, then ac < bc.
This time, let c = -10
Property 4 implies that when both sides of an inequality are
multiplied by a negative number, the direction of the symbol
must be reversed.

6. Ex: 1 Find all real numbers satisfying 2x – 1 < 4x + 3.
Isolate x.
Don’t forget property 4 if you subtract 4x from both sides.
Solution: -2 < x x -2

7. Ex: 2 Find all real numbers x satisfying
Solution: You can solve this as one inequality or break it
up into two inequalities.
(Now solve both)
-2 < x
Now, write the solution as one inequality. x 1 -2

8. Intervals: Representing sets of numbers (table p. 7)
open interval
closed interval
half open interval
half open interval
unbounded above
unbounded above
unbounded below
unbounded below

9. 3.) Find all the values of x satisfying the inequality
Solution:
Set one side equal to zero:
Factor:
(x – 3)(x – 1) > 0
Use a sign graph to determine where on the
coordinate line this holds true.
(x – 3)
(x – 1)
(x – 3)(x – 1) o o 1 3
Since our inequality is “greater than” zero we want the positives.
The values of x that are satisfied lie in that area.

10. Ex: 4 Find all values of x that satisfy
(5 – x)
(x – 1)
(x – 3)
(5 – x)(x – 1)(x – 3) 1 3 5
Since the inequality is “less than or equal to zero” we want the
negatives.
Our answer written in interval notation:

11. o o Ex: Find all values of x for which x+1 x-1 x 2-x fcn o o x -1 1 2
We want the positives and the endpoints unless it would make
the denominator “0”.
Solution:

12. Absolute Values:
The absolute value of a real number x, denoted |x|, describes
the distance on the coordinate line from the number x to the
number 0.
Ex: 6 Determine all values of x for which
Recall: When you take the absolute value of a number or an
expression, the result is positive. However, we do not know
whether the original number or expression was positive or negative.
We must consider both possibilities.
Notice the absolute
value signs are now
gone.

13. Solve for x
x – 2 = 3(2x + 1)
-3(2x+1) = x – 2
x- 2 = 6x + 3
-6x – 3 = x - 2
-2 = 5x + 3
-3 = 7x - 2
-1 = 7x
-5 = 5x
-1/7 = x or
-1 = x

14. Distance formula:
The distance between two numbers on a coordinate line can be found by
We use absolute value because distance can never
be negative.
Midpoint formula:
We can find the midpoint between two
numbers on a coordinate line by
Note: It does not matter which number you let be x1 or x2

15. Ex: 7 Find all values of x that satisfy |2x – 1| < 3
This is just like absolute values involving equations. You
must make two inequalities to account for both possibilities.
-3 < 2x – 1 < DO NOT CHANGE THE DIRECTION
OF THE INEQUALITY SYMBOL!!!
-3 < 2x – 1 and 2x – 1 < 3
-2 < 2x x < 4
-1 < x and x < 2
O O