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