Removing grouping symbols: Level 2
The relationship of a − b to b − a
We know the relationship of a + b to b + a:
a + b = b + a.
But what is the relationship of a − b to b − a ?
a − b is the negative of b − a.
a − b = −(b − a).
For example, 2 − 5 = −(5 − 2).
Problem 20. Prove: a − b = −(b − a).
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| Solution 1. | a − b | = | a + (−b) |
| = | −b + a | ||
| = | −(b − a). | ||
Solution 2. Two numbers are negatives of one another if their sum is 0.
| If | |||
| p + q | = | 0, | |
| then | |||
| p | = | −q. | |
a − b and b − a clearly satisfy that test:
a − b + b − a = 0.
Therefore, a − b = −(b − a).
| Problem 21. | x − 2 2 − x |
= | −1 |
| The fraction has the form: | a −a |
, which for all a (except 0) is equal |
to −1.
| The rule: | −a −b |
= | a b |
We may interpret that rule (Lesson 4) to mean:
In any fraction we may change the signs
in both the numerator and denominator.
| Example. Apply that rule to | p − q −2 |
. |
| Answer. | p − q −2 |
= | q − p 2 |
q − p is the negative of p − q.
Problem 22. Apply that rule.
| a) | a − b −c |
= |
b − a c |
b) | 2 − x −5 |
= |
x − 2 5 |
| c) | −x + y − z −4 |
= |
x − y + z 4 |
d) | −x − 1 2 |
= |
x + 1 −2 |
= | − |
x + 1 2 |
Next Lesson: Adding like terms
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