Inequalities

As with an equation, we solve an inequality in the variable X by finding all values of X for which the inequality is true. Such values are called solutions and we say that the solutions satisfy the inequality. The set of all real numbers that are solutions of an inequality is called the solution set of the inequality. The procedures for solving linear inequalities in one variable are much like those for solving linear equations. To isolate the variable we make use of properties of inequalities which are similar to the properties of equality. However, there is an important exception. WHEN BOTH SIDES OF AN INEQUALITY ARE MULTIPLIED OR DIVIDED BY A NEGATIVE NUMBER, THE DIRECTION OF THE INEQUALITY SYMBOL MUST BE REVERSED. Solve the linear inequality and sketch the solution set on the real number line.

Example

Solve the linear inequality and sketch the solution set on the real number line.

Solution

A.

2X + 3 > 4

2X > 1

X > 1/2

B.

3X - 4 < 2x - 5

x < 9

C.

- 5X + 4 > 3

- 5X > -1

X < 1/5

To solve a quadratic inequality, we use the fact that a polynomial can change signs only at its zeros (the x-values that make the polynomial zero). Between two consecutive zeros a polynomial must be entirely positive or entirely negative. This means that when the real zeros of a polynomial are put in order, they divide the real line into intervals in which the polynomial has no sign changes. We call these zeros the critical numbers of the inequality and we call the resulting intervals the test intervals for the inequality. To determine the intervals on which the values of a polynomial are entirely negative or entirely positive, use the following steps.

1. Find all the real zeros of the polynomial, and arrange the zeros in increasing order (from smallest to largest). The zeros of the polynomial are its critical numbers.

2. Use the critical numbers of the polynomial to determine its test intervals.

3. Choose one representative x-value in each test interval and evaluate the polynomial at that value. If the value of the polynomial is negative, then the polynomial will have negative values for every x-value in that interval. If the value of the polynomial is positive, then the polynomial will have positive values for every x-value in that interval.

The concepts of critical numbers and test intervals can be extended to inequalities involving rational expressions. To do this, we use the fact that the value of a rational expression can change signs only at its zeros (the x-value for which its numerator is zero) and its undefined values (the xvalues for which its denominator is zero). These two types of numbers make up the critical numbers of a rational inequality.