# Variables, Expressions, Equations, Inequalities

## Variables

Variable - a symbol that can stand for an unknown number or for a quantity that changes

Variables are often represented by letters.

## Expressions

Expression - a representation of some number

• Made up of numbers, variables, operations, and grouping symbols

• Does not have an equal symbol

Order of operations (PEMDAS) - used to simplify expressions

• P: parenthesis (grouping symbols) - things inside parenthesis are to be done first, and if there are multiple operations in the parenthesis, apply PEMDAS

• E (exponents) is something you will learn later

• M/D: multiplication/division is to be done left to right

• A/S: addition/subtraction is to be done left to right

### Example

What is 5 × (3 + 2) × 2 + 4 - 2?

PEMDAS

There are parentheses, so we evaluate that first.

5 × 5 × 2 + 4 - 2

Next, we do multiplication/division from left to right.

25 × 2 + 4 - 2

50 + 4 - 2

Finally, we do addition/subtraction from left to right.

54 - 2

52

Algebraic expression - an expression that contains variables

• Numbers are used when quantities are known; variables are used when the quantities are unknown

• For example, if b stands for the number of doughnuts in one full box, “the number of doughnuts in a full box and four extra doughnuts” can be represented by b + 4

### Example

If c stands for the number of chips in a bag of chips, write an expression that represents the number of chips in five bags.

Since one bag has c chips and we want the number of chips in 5 bags, we need to multiply the number of chips in one bag by the number of bags: c × 5 = 5c.

## Equations

Equation - sets two quantities as equal (=)

• An example of an equation is 3 + 3 = 6

• Variables can also be used in equations: h - 9 = 63

• Problem situations can be represented by equations

• For example, if b stands for the number of doughnuts in one full box, “two full boxes of doughnuts contain a total of 36 doughnuts” can be represented by 2b = 36

### Example

If a stands for the number of apples in a box, write an equation to represent the following statement: “one box of apples and 5 extra apples in total is 82 apples”.

One box of apples + 5 more apples = 82 apples

Since one box of apples can be represented by a, we can put that into the equation:

a + 5 = 82

## Inequalities

Inequality - compares two quantities and shows whether one is greater than, equal to, or less than the other

Inequality signs:

• < means the quantity on the left is less than the one on the right

• > means the quantity on the left is greater than the one on the right

• ≤ means the quantity on the left is less than or equal to the one on the right

• ≥ means the quantity on the left is greater than or equal to the one on the right

Inequalities can have variables, just like equations.

Some examples of inequalities:

• 3 < 6

• 4 ≤ 4

• 7 ≥ 5

• p > 10

• q + 3 ≤ 7

### Example

If s represents the number of students in a classroom, write an inequality for this statement: twice the number of students in the classroom is less than or equal to 49 students.

Twice the number of students ≤ 49

The number of students is s and twice of something means 2 times:

2s ≤ 49