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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
The direction of the inequality symbol changes when both sides of the inequality are multiplied or divided by a negative number. For example, in the inequality -2x > 4, the solution becomes x < -2 after dividing by -2.

Step by step solution

01

Understanding Inequality Rules

The first thing that should be clear is when we are dealing with inequalities, the direction of the inequality sign ('<', '>') is important and represents the relationship between the values on either side of the sign. The sign indicates that one value is less than or greater than the other.
02

Conditions for Inequality Sign Reversal

In simple terms, whenever we multiply or divide both sides of the inequality by a negative number, the direction of the inequality sign changes. This is because of the rules of order in arithmetic, where the relationship between negative numbers is 'reversed' compared to that of positive numbers. For example, -3 is less than -2, whereas in positive numbers, 3 is more than 2.
03

Example of Inequality Sign Reversal

Let's consider an example. If we have the inequality -2x > 4, to solve this inequality for x, we need to divide both sides by -2. The correct way to write the solution is as x < -2, not as x > -2. The change in the direction of the inequality sign occurs because we divided by a negative number.

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