We discuss inequalities.

*exactly*82. It may have been more advantageous to ask what score was needed to have an average of at least 82, yielding not an equation, but an

*inequality*, .

As with equations, there are various types of inequalities.

A *linear inequality in * is an inequality which is equivalent to , , , or . We solve
the inequality in much the same manner as with a linear equation. The
main differences come from changing the direction of the inequality when
multiplying/dividing by a negative quantity and expressing our answers in interval
notation.

Nonlinear inequalities are more complicated. To solve them, we will use a tool called
a *sign chart*. The process requires us to move all the nonzero terms of the inequality
to one side, and factor.

Notice where the left-hand side would be either zero or undefined. Those occur at . We’ll use those points to split the number line into four regions. Inside each of those regions, our factors are either always positive or always negative.

For , we find , , and . Together, that means the left-hand side of our inequality has the form . It is, therefore, positive in that region, and makes up part of our solution. We cannot include the endpoint in the solution, as the fraction is undefined there.

In the same way, we see that in the intervals and , the left-hand side of the inequality is also positive. Notice that the inequality is NOT satisfied at , where the left-hand side equals zero.

The solution is .