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MAT091 :: Lecture Note :: Week 04
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### Intervals

An interval is a range of values that start with a minimum and end with a maximum.

Intervals can be represented using the following notation.

```	[min, max]  where [ implies min is in the interval
and ] implies max is in the interval

(min, max]  where ( implies min is not in the interval
and ] implies max is in the interval

[min, max)  where [ implies min is in the interval
and ) implies max is not in the interval

(min, max)  where ( implies min is not in the interval
and ) implies max is not in the interval

brackets imply end points are included
parenthesis imply end points are excluded
```

Examples.

```	[0, 5]  implies all numbers between and including 0 and 5
(0, 5]  implies all numbers greater than 0 and less than or equal 5
```

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### Algebraic Expressions

An algebraic expression is a mathematical expresssion that contains at least one variable along with zero or more numbers (i.e. constants) and zero or more arithmetic operators.

The following are algebraic expressions.

```   n
3n
5n + 10
2x^2 - 2y^3
9a + 2b - 3c
11(x + y^-3 + 2w)
```

Variables are unknown values that are represented by single-character letters. For example, `3n` is `3 times 'n'`, where 'n' is a variable. If a value is assigned to 'n', then the expression can be evaluated.

```   3n + 8

if n = 3, then 3(3) + 8 = 100
if n = -5, then 3(-5) + 8 = -7
if n = 0, then 3(0) + 8 = 8
```

The replacement (or swapping) of variables with values can be described in a variety of ways: Values can be "plugged into" or "assigned to" variables; or variables can be "replaced with" or "substituted with" values; or we "let" variables be certain values.

```   5x + 1  assign 5 to x ........... 5(5) + 1 = 25 + 1 = 26
5x + 1  replace x with 5 ........ 5(5) + 1 = 25 + 1 = 26
5x + 1  plug 5 into x ........... 5(5) + 1 = 25 + 1 = 26
5x + 1  substitute x with 5 ..... 5(5) + 1 = 25 + 1 = 26
5x + 1  let x be 5 .............. 5(5) + 1 = 25 + 1 = 26
```

Again, expressions are typically evaluated after variables have been assigned (i.e. given) values.

If a variable is multiplied by a constant, then the constant is called a coefficient.

```   3n ...... 'n' is a variable and 3 is a coefficient
x ....... 'x' is a variable and 1 is the coefficient
-1y ..... 'y' is a variable and -1 is the coefficient
```

The combination of a coefficent and variable is called a factor. Factors are numbers, variables, and expressions that are multiplied together to produce a product.

Factors separated by addition and subtraction operators are called terms.

```   3n + 1 .............. 1 factor, 2 terms
4a + 2b - 10 ........ 2 factors (4a and 2b), 3 terms
7x - 5 + 8y - 11 .... 2 factors (7x and 8y), 4 terms
2x^3 + 10^2.......... 1 factor (2x^3), 2 terms
3(4) - 2(1 + 3)...... 2 factors, 2 terms
```

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