Other Important Math Concepts

Commonly in algebra and other upper-level fields of math you find yourself working with mathematical expressions that do not equal each other. The statement comparing such expressions with symbols such as < (less than) or > (greater than) is called an inequality.

An example of an inequality is .

To solve for , simply divide both sides by  and the solution is shown to be .

Graphs of the solution set of inequalities are represented on a number line.

Open circles are used to show that an expression approaches a number but is never quite equal to that number.

Conditional inequalities are those with certain values for the variable that will make the condition true and other values for the variable where the condition will be false.

Absolute inequalities can have any real number as the value for the variable to make the condition true, while there is no real number value for the variable that will make the condition false.

Solving inequalities is done by following the same rules as for solving equations with the exception that when multiplying or dividing by a negative number the direction of the inequality sign must be flipped or reversed.

Double inequalities are situations where two inequality statements apply to the same variable expression.

An example of this is
.

A weighted mean, or weighted average, is a mean that uses 'weighted' values. The formula is
.

Weighted values, such as
 are assigned to each member of the set
.

If calculating weighted mean, make sure a weight value for each member of the set is used.

A fraction that contains a fraction in the numerator, denominator, or both is called a complex fraction. These can be solved in a number of ways; with the simplest being by following the order of operations as stated earlier. For example,
.

Another way to solve this problem is to multiply the fraction in the numerator by the recipricol of the fraction in the denominator. For example,
.

In order to solve a radical equation, begin by isolating the radical term on one side of the equation, and move all other terms to the other side of the equation. Look at the index of the radicand.

Remember, if no number is given, the index is 2, meaning square root. Raise both sides of the equation to the power equal to the index of the radical. Solve the resulting equation as you would a normal polynomial equation. When you have found the roots, you must check them in the original problem to eliminate extraneous roots. The solution set is the set of all solutions of an equation. Many equations will only have one value in their solution set. If there were more solutions then they would also be included in the solution set. When an equation has no true solutions, this is referred to as an empty set.