Expression: (no = sign)can be simplified or factored,

but NOT solved

Equation: two equal expressions (has = sign)

CAN be solved

Equations:Conditional Equation: finite

solution setx2 – x – 6 = 0

Solution Set: { –2, 3}Identity: variable can be any real

number2(x – 3) = x – 6 + x

Solution Set: {reals}

Number Sets:Counting or Natural #s:

{1, 2, 3, 4, ...}Whole #s:

{0, 1, 2, 3, 4, ...}Integers:

{...,-3, -2, -1, 0, 1, 2, 3, 4, ...}

Definition:Rational #: can be written as the

ratio of two integers where b ≠ 0.

That is: integers, fractions, terminating & repeating decimals.

Recall

ab

1 0.33

Definition:Irrational #: a real number that is

not rational. (Duh!) That is: non-terminating, non-

repeating decimals like:0.12345678910111213141...

0.12122122212222..., ,, 2 5 e 2.718...

Degree of an expression or equation:

The greatest power on any one term 5x7 + 11x5 – 7x3 + 2x (7th degree)

OR The greatest SUM of powers on any one term

5x2y3 + 11x2y7 – 7xy3 + 2 (9th degree)

Disjunction: “Or” statement – Take the union of two solution

sets!

2x + 5 < 3 or 1 – 2x < 7

2x < – 2 – 2x < 6 x < – 1 OR x > 3

Solution Set: {x: x < – 1 or x > 3}

Conjunction: “And” statement – Take the intersection of two

solution sets!

– 11< 2x + 5 <1 3– 11< 2x + 5 and 2x + 5 <1 3

– 16 < 2x 2x < 8 –8 < x AND x < 4

Solution Set: {x: – 8 < x < 4}

Absolute Value

Absolute Value of a real number is the distance to the origin on the real number line.

Formal Definition:

00 0

0

x if xx if x

x if x

The distance between two numbers a and b uses absolute

value because we can subtract in either order and then make the

answer positive(distances are never negative).

e.g. Distance between 4 and -12 is|4 – -12| or |-12 – 4| |16| or |-16| 16

Formal Definition of Distance between two real numbers:

The distance between a and b is given by the absolute value of the

difference of the coordinates.

Distance between a and b = |a – b| or |b – a|

Check your understanding:

T F 1. |a| > 0T F 2. |a2| = a2

T F 3. |a3| = a3 T F 4. |a + b| = |a| + |b|T F 5. |ab| = |a| . |b|

Check your understanding:F 1. |a| > 0 (could be zero)T 2. |a2| = a2 (always non-negative)F 3. |a3| = a3 (not when a < 0)F 4. |a + b| = |a| + |b| (e.g. when a > 0 and b < 0)T 5. |ab| = |a| . |b| (makes it positive sooner or later)

Absolute value equations may have zero, one or TWO

solutions:

Example 1:|a + 5| = 15

Absolute value equations may have zero, one or TWO

solutions:Example 1:

|a + 5| = 15

a + 5 = 15 OR a + 5 = -15solution set: {10, -20}

Absolute value equations may have zero, ONE or two

solutions:

Example 2: |a – 7.2| = 0

Absolute value equations may have zero, ONE or two

solutions:

Example 2: |a – 7.2| = 0

a = 7.2 {7.2}

Absolute value equations may have ZERO, one or two

solutions:

Example 3: |3a - 2| = -5

Absolute value equations may have ZERO, one or two

solutions:

Example 3: |3a - 2| = -5

Absolute value equations: Check your understanding.

Example 4:

4 33x

Absolute value equations: Check your understanding.Example 4:

4 33x

{-21, -3}

4 3 4 33 3x xor

7 13 3x xor

21 3x or x

Absolute Value Inequalities:Think: Is the solution of | x | > 11 a

disjunction or a conjunction?

Think: Is the solution of | x | ≤ 3 a disjunction or a conjunction?

-3 30

-11 110

Absolute Value Inequalities:1. Isolate the abs value sign on one

side of the equation.2. Separate into a disjunction or a

conjunction of two statements.3. Solve each statement alone.4. Combine to find the disjunction

or conjunction.

Absolute Value Inequalities:

Example 1: | x + 2 | + 4 < 11

Isolate abs value first: | x + 2 | < 7

Absolute Value Inequalities:

Example 1: | x + 2 | < 7

Begin by imagining: distance of some expression is less than 7 from origin!

-7 70

Absolute Value Inequalities:

Example 1: | x + 2 | < 7

2. Separate into a disjunction or a conjunction of two statements

x + 2 > - 7 AND x + 2 < 7

-7 70

Absolute Value Inequalities:

Example 1:| x + 2 | < 7

x + 2 > - 7 AND x + 2 < 7 x > -9 AND x < 5

3. Solve each statement alone.4. Combine to find the disjunction or conjunction.

Absolute Value Inequalities:

Example 1:| x + 2 | < 7

x + 2 > - 7 AND x + 2 < 7 x > -9 AND x < 5

{x: -9 < x < 5}*Hint: Abs Value < Pos # became a

CONJUNCTION

Absolute Value Inequalities:

Example 2: | 3x - 5 | > 2

Begin by picturing: distance is more than 2 units from the origin!

-2 20

Absolute Value Inequalities:

Example 2: | 3x - 5 | > 2

2. Separate into a disjunction or a conjunction of two statements

3x - 5 < - 2 OR 3x - 5 > 2

-2 20

Absolute Value Inequalities:

Example 2: | 3x - 5 | > 23x - 5 < - 2 OR 3x - 5 > 2 3x < 3 OR 3x >7

3. Solve each statement alone.4. Combine to find the disjunction or conjunction.

Absolute Value Inequalities:

Example 2: | 3x - 5 | > 23x - 5 < - 2 OR 3x - 5 > 2 3x < 3 OR 3x >7

{x: x<1 or x > }73

Absolute Value Inequalities:

Example 2: | 3x - 5 | > 23x - 5 < - 2 OR 3x - 5 > 2

3x < 3 OR 3x >7

{x: x<1 or x > }*Hint: Abs value > pos # became

a DISJUNCTION!

73

Absolute value inequalities: Check your understanding.

Example 3:

2 4 4 23

x

Absolute value inequalities: Check your understanding.

Example 3: 2 4 4 23

x

4 33

4 3 4 33 3

7 13 3

21 3: 21 3

x

x xor

x xor

x or xx x or x

Absolute value inequalities: Watch for special cases.

Example 4:

4 3x

Absolute value inequalities: Watch for special cases.

Example 4: {real numbers}

*Absolute values are ALWAYS at least zero!

4 3x

Absolute value inequalities: Watch for special cases.

Example 5: 4 3x

Absolute value inequalities: Watch for special cases!

Example 5:

*Absolute values canNEVER be less than zero!

4 3x