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Answer:
1. a+c is larger than b+d
2. No way to tell whether a+d or b+c is larger.
Step-by-step explanation:
<u>1. Which is larger, a+c or b+d?</u>
Let a, b, c, and d be any numbers such that .
Specifically, note that , and subtracting b from both sides of the inequality, observe that
.
Similarly, , and subtracting d from both sides of the inequality, observe that
.
From this, <u>add "a-b"</u> (a positive number, as proven above) to both sides of the inequality.
Addition by zero (<u>the additive identity</u>) doesn't change anything, so the right side remains "a-b"...
... and <u>"a-b" is positive</u>...
... so, by the <u>transitive property</u> of inequality...
Recall that <u>subtraction is addition by a negative</u> number...
...and that <u>addition is associative and commutative</u>, so things can be added in any order, so the middle two terms on the left side can be rearranged...
<u>Adding b + d</u> to both sides of the inequality
... and <u>simplifying</u>
So, a+c is larger than b+d.
<u>2. Which is larger, a+d or b+c?</u>
Consider the following two examples:
<u>Example 1</u>
Suppose a=10; b=3; c=2; d=1.
Note that (
) and, also observe that
, and
, so a+d is larger than b+c.
<u>Example 2</u>
However, suppose a=10; b=9; c=8; d=1.
Note that (
) but that
, and
, so a+d is smaller than b+c.
So, in one example, a+d is bigger, and in the other, a+d is smaller. Therefore, there is no way to tell which of a+d or b+c is larger from only the given information.