Problem 114 - Introductory Algebra for College Students

Question

When solving an inequality, when is it necessary to change the direction of the inequality symbol? Give an example.

Step-by-Step Solution

Answer

When solving an inequality, the direction of the inequality symbol should be changed when either multiplying or dividing by a negative number. For instance, in the inequality -3x > 6, when divided by -3, the inequality symbol changes direction to become x < -2.

Step 1: Understanding Rules of Inequalities

An essential rule in solving inequalities is that whenever the expressions on either side are multiplied or divided by a negative number, the direction of the inequality symbol should be reversed. This is because the value that was once greater than (or less than), after multiplication or division by a negative number, immediately becomes less than (or greater than). Therefore, this rule helps maintain the integrity of the relationship expressed by the inequality.

Step 2: Example of Changing Inequality Symbol Direction

Consider an inequality -3x > 6. To solve for x, both sides must be divided by -3. But, in doing so, the inequality's direction must be reversed. Hence, the solution of the inequality becomes x < -2.

Step 3: Justification

To confirm, if one inserts x < -2 back into the original inequality -3x > 6: -3*(-3) > 6 which simplifies to 9 > 6, a true statement, confirms the solution. However, if the inequality's direction were not changed, we would have x > -2. Inserting x > -2 back into the original inequality gives -3*(-1) > 6, shrinking down to -3 > 6, a false statement, thus verifying the necessity of switching the inequality's direction whenever dividing or multiplying by a negative number.