1. 1 Preliminaries Precalculus Review I Precalculus Review II
The Cartesian Coordinate System
Straight Lines

2. Learning Objectives
Review elementary mathematics
Know how to express mathematics in English
Understand some terminology

3. Arithmetic Symbols
＋ Add (plus), Addition, Sum
－ Subtract (minus), Subtraction, Difference
× Multiply (time), Multiplication, Product
÷ Divide, Division, Quotient

4. The Real Numbers
The real numbers can be ordered and represented in order on a number line
-1.87
4.55 x

5. Inequalities, graphs, and intervals
Inequality
Graph
Interval
( ] ( 5 ]
) or ( means not included in the solution
] or [ means included in the solution

6. Intervals Interval Graph Example (a, b) [a, b] (a, b] [a, b) (a, )
( )
[ ]
( ]
[ ) ( ) [ ]
(3, 5)
[4, 7]
(-1, 3]
[-2, 0)
(1, )
(- , 2)
[0, )
(- , -3]
( )
[ ]
( ]
[ ) ( ) [ ]
a b
a b
a b a 1 b 2 a b -3

7. Properties of Inequalities
Example
If a, b, and c are any real numbers, then
Property 1
Property 2
Property 3
Property 4
2 < 3 and 3 < 8, so 2 < 8.

8. Absolute Value
Notice the opposite sign
To evaluate:

9. Absolute Value Properties
If a and b are any real numbers, then
Property 5
Property 6
Property 7
Property 8
Example

10. Exponents
n,m positive integers
Definition
Example
n factors

11. Laws of Exponents
Law
Example

12. Algebraic Expressions
Polynomials
Rational Expressions
Other Algebraic Fractions

13. Polynomials Addition Subtraction Combine like terms Distribute

14. Polynomials
Multiplication
Distribute
Distribute
Combine like terms

15. Factoring Polynomials
Greatest Common Factor
The terms have 6t2 in common
Grouping
Factor mx
Factor –2

16. Factoring Polynomials
Difference of Two Squares:
Ex.
Sum/Difference of Two Cubes:
Ex.

17. Factoring Polynomials
Trinomials
Ex.
Trial and Error
Ex.
Greatest Common Factor
Trial and Error

18. Roots of Polynomials Finding roots by factoring
(find where the polynomial = 0)
Ex.

19. Roots of Polynomials Finding roots by the Quadratic FormulaIf
with a, b, and c real numbers, then

20. Example Using the Quadratic Formula: Ex. Find the roots of
Note values
Here a = 3, b = 7, and c = 1
Plug in
Simplify

21. Rational Expressions Operation P, Q, R, and S are polynomials Addition
Notice the common denominator
Subtraction
Find the reciprocal and multiply
Multiplication
Division

22. Rational Expressions Simplifying Multiplying Cancel common factors2
Multiply Across

23. Rational Expressions Adding/Subtracting Must have LCD: x(x + 4)
Combine like terms
Distribute and combine fractions

24. Other Algebraic Fractions
Complex Fractions
Multiply by the LCD: x
Distribute and reduce to get here
Factor to get here

25. Other Algebraic Fractions
Notice:
Rationalizing a Denominator
Multiply by the conjugate
Simplify

26. Cartesian Coordinate System
y-axis
(x, y)
x-axis

27. Cartesian Coordinate System
Ex. Plot (4, 2)
Ex. Plot (-2, -1)
Ex. Plot (2, -3)
(4, 2)
(-2, -1)
(2, -3)

28. The Distance Formula

29. The Distance Formula Ex. Find the distance between (7, 5) and (-3, -2)10

30. The Equation of a Circle
A circle with center (h, k) and radius of length r can be expressed in the form:
Ex. Find an equation of the circle with center at (4, 0) and radius of length 3

31. Straight Lines
Slope
Point-Slope Form
Slope-Intercept Form

32. Slope – the slope of a non-vertical line that passes through the points is given by:
and
Ex. Find the slope of the line that passes through the points (4,0) and (6, -3)

33. Slope
Two lines are parallel if and only if their slopes are equal or both undefined
Two lines are perpendicular if and only if the product of their slopes is –1. That is, one slope is the negative reciprocal of the other slope (ex ).

34. Point-Slope Form
An equation of a line that passes through the point with slope m is given by:
Ex. Find an equation of the line that passes through (3,1) and has slope m = 4.

35. Slope-Intercept Form
An equation of a line with slope m and y-intercept is given by:
Ex. Find an equation of the line that passes through (0,-4)
and has slope

36. Vertical Lines y
Can be expressed in the form x = a
x = 3 x

37. Horizontal Lines y
Can be expressed in the form y = b
y = 2 x

38. Example
Find an equation of the line that passes through (-2, 1) and is perpendicular to the line
Solution:
Step 1.
Step 2.

39. Example
Find an equation of the line that passes through (0, 1) and is parallel to the line
Solution:
Step 1.
Step 2.