Maintaining Mathematical Proficiency
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Consider some specific examples on the number line.
- b < - a
We know that a number a is to the left of a number b on the number line. This means that a is less than b, which we can write as a
For our first specific example, we can let a and b equal any values such that a lies to the left of b. We will arbitrarily choose two positive numbers, a=2 and b=8. This gives the following number line.
Since - b lies to the left of - a, we have that - b<- a.
Let's now arbitrarily choose two negative numbers, a=- 10 and b=- 4. This gives the following number line.
If a=- 10 and b=- 4, then we have - a = 10 and - b= 4.
Since - b lies to the left of - a, we again obtain that - b<- a.
Finally we will consider the last possible option. We will arbitrarily choose one negative number, a=- 1, and one positive number, b=3. This gives the following number line.
If a=1 and b=6, then we have - a = - 1 and - b= - 6.
Since - b lies to the left of - a, for the third time we get that - b<- a.
Although we showed only three examples, this reasoning will hold true for all possible values of a and b. Therefore, we obtained that if a
Multiply by -1 and flip inequality sign
Rearrange inequality
Once again, we can see that - b is less than - a.