An ** equation** expresses the equality of two mathematical expressions.

An equation always contains an equal sign.

A ** solution** of an equation is a number that, when substituted for the variable, results in a true equation.

To ** solve** an equation is to find all solutions for the equation.

** Equivalent equations** are equations that have exactly the same solutions. When solving equations, we usually try to find equivalent equations until we get to

A few things we can do that result in equivalent equations:

- Add numbers to both sides of an equation
- Multiply both sides by a nonzero number
- Replace things that are equal with each other

If you need more practice with these, here is a worksheet on solving multi-step linear equations.

# Contradictions, Conditional Equations and Identities

An equation that has no solutions is called a **contradiction**.

An equation that is true for some values of the variable, but not true for others is called a **conditional equation**.

An **identity** is an equation that is true for all values of the variable for which all terms of the equation are defined. It has an infinite number of solutions.

# Absolute Value Equations

The absolute value of a number is the “non-negative” version of that number. The absolute value of is written

So the absolute value of -7 is 7 and the absolute value of 9 is 9. The absolute value of 0 is 0.

Formally, |x| = x if and |x|=-x (which is nonnegative) if

Here is a worksheet for extra problems on absolute value equations.