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Chapter 2: Problem 114

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.

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