Why does the inequality sign change when both sides are multiplied or divided by a negative number?
suppose we start with an example
10 > 4 .....true
multiply by 2
20 > 8 ... still true
multiply by -2
-20 > - 8 .... false
how do we make it true???
-20 < -8 ... now true!
Conclusion???
Multiply by -1 on both sides?
The inequality sign changes when both sides of an inequality are multiplied or divided by a negative number because negative numbers have the property of changing the order of numbers when multiplied or divided. To understand this, let's consider a simple example.
Suppose we have the inequality 5 > 2.
If we multiply both sides of the inequality by -1, we get: -5 < -2.
Here's why the inequality sign changes:
1. Multiplication: When we multiply both sides by -1, we essentially flip the signs of the numbers. So, on the left side, 5 becomes -5, and on the right side, 2 becomes -2.
2. Order: Negative numbers have a different order compared to positive numbers. -5 is smaller than -2, so the inequality changes from "greater than" to "less than."
In general, if you multiply or divide an inequality by a negative number, the inequality sign will change direction. This is because negative numbers introduce a reversal of the order of numbers. This property is important to note when manipulating inequalities in algebraic expressions.