Rules for reversing the inequality sign

Reversing the Inequality Sign: An Overview

Inequality equations are similar to regular equations except they use inequality signs (> , < , ≥ , ≤) instead of an equals sign.
The rules for reversing the inequality sign are required when we modify the inequality during the process of solving it.
These rules are fundamental in algebra and often come into play in various mathematical calculations.

Basic Rules for Reversing the Inequality Sign

When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
For instance, if you have ‘-2x < 4’ and you want to isolate ‘x’, you would divide both sides by ‘-2’, reversing the sign in the process, resulting in ‘x > -2’.
This rule applies whether the variables, coefficients, or constants are positive or negative.

Special Cases for Reversing the Inequality Sign

There are particular instances where reversing the inequality sign varies, mostly depending on operations performed:
- When adding or subtracting a number on both sides of an inequality, you do not reverse the inequality sign.
- For example, ‘x - 5 < 7’ can be transformed into ‘x < 7 + 5’ without switching the inequality sign, resulting in ‘x < 12’.
- However, when multiplying or dividing by a variable in inequalities, we do not know the value of the variable; it could be positive or negative. In this situation, we can’t simplify the inequality by multiplying or dividing unless we can confirm the sign of the variable.

Examples of Reversing Inequality Sign

Consider the inequality ‘-3x > 6’. To get ‘x’ alone, you divide both sides by ‘-3’, yielding ‘x < -2’. This example illustrates the reversing of the inequality sign.
Without reversing the inequality sign, we would mistakenly write ‘x > -2’, which is incorrect.
However, for ‘x + 2 < 5’, solving for ‘x’ means subtracting 2 from both sides, which gives ‘x < 3’. The inequality sign doesn’t need to be reversed in this instance.

Understanding the Reason for the Rule

Reversing the inequality sign when multiplying or dividing by a negative number is crucial for maintaining the truth of the inequality.
The reason is that negative numbers, by definition, fall to the left on the number line, which is the opposite direction to positive numbers. Therefore, when we multiply by a negative number, we reverse the order of the numbers, hence the need to reverse the inequality sign.

Conclusion

The rules for reversing the inequality sign in algebra can seem tricky at first, but with practice, they become intuitive.
Always bear in mind to reverse the inequality sign when multiplying or dividing by a negative number.
These rules are essential for correctly solving and interpreting inequalities in algebra and other areas of mathematics.