Application of Nonlinear Inequalities

A nonlinear inequality describes a comparison involving an expression that curves or behaves more complexly than a straight line. These inequalities often appear in forms that include squares, products, or variables in the denominator.

To solve such an inequality, one starts by rewriting it so that zero appears on one side. For example, the inequality:

can be factored as:

This form makes it easier to identify the values that cause the expression to equal zero. In this case, the key values are 3 and -2. These values divide the number line into three regions.

The solver chooses one sample value in each region and substitutes it into the expression. This test shows whether the expression within that section is positive or negative. If the original inequality requires the expression to be less than zero, the correct regions are those where the result is negative. The complete solution includes only those regions.

The same approach applies to rational inequalities, where the variable appears in the denominator. However, the solver must also consider where the expression is undefined. For example:

In this case, the key values are -1 and 4. These divide the number line into separate intervals. The solver tests each interval and excludes the point where the denominator becomes zero—in this case, 4.

Nonlinear inequalities are useful in practical settings. A business analyst might use one to determine when profit stays above zero. An engineer might apply one to ensure a machine operates within safe pressure limits. A physicist might analyze when an object stays within a certain range of motion.