Absolute Value Equations

Absolute value, by definition, is the distance from zero on a number line.

How to Solve Absolute Value Equations Playlist

If we have |x| = 3 then the question is, “what number can x be so that the distance to zero is 3?” There are two solutions to this question -3 and 3.

In general,

Here n is a positive integer and X represents an algebraic expression called the argument of the absolute value. To solve these equations, set the argument equal to plus or minus n and solve the resulting equations.

Solve.

This technique requires us to first isolate the absolute value. Apply the usual steps for solving to obtain the absolute value alone on one side of the equation, and then set the argument to plus or minus n.

Solve.

When two absolute values are equal we can set the argument of one equal to the argument of the other.

Tip: When in doubt of a solution, check to see if it solves the original equation.  For example, check that {-7, 3} is the solution set for,

Notice that both numbers solve the original equation and therefore we have verified they are in the solution set.

YouTube Videos:

Linear Inequalities (one variable)

Solving Linear Inequalities Playlist on YouTube

Solve and graph the solution set. 

(I.N. stands for interval notation here.)

Compound inequalities can be split up or solved in one step like the following examples. Note that all inequality symbols face the same direction when combined.

Average Problem: Clint wishes to earn a B which is an average of at least 80 but not more than 90.  What range must he score on the fourth exam if the first three were 65, 75, and 90?

Commission Problem: Bill earns $12.00 plus $0.25 for every person he gets to register to vote. How many people must he register to earn at least $50.00 for the day?

Video Examples on YouTube: