Inequality Multiplication By Negative Value

This property allows us to multiply a value to both sides of the operator. But when the value is negative, we need to flip the operator. Thus, the less-than operator becomes the greater-than operator, and vice versa.

Here is a visual representation of this property:

On the right side of the graph, the orange bar extends to 3, while the blue bar extends to 6. We can see that the orange bar is shorter than the blue bar.

The left side shows what happens when we multiply by -1. The orange bar extends to -3, while the blue bar extends to -6. Notice that -3 is greater than -6. The orange bar is still shorter, but that means that the orange bar is less negative than the blue bar.

Here are some examples:

1. if 3 < 5, then 3 ⋅ (-5) > 7 ⋅ (-5)

2. if x < 10, then x ⋅ (-8) > 10 ⋅ (-8)

3. if 3 ⋅ m < 15 + n, then (3 ⋅ m) ⋅ (-1) > (15 + n) ⋅ (-1)

If we don't know whether the value is positive or negative, then we do not know whether to flip the operator or not, so we cannot draw a conclusion.

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