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Sagot :
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:
1. Which is larger, a+c or b+d?
Let a, b, c, and d be any numbers such that [tex]a > b > c > d[/tex].
Specifically, note that [tex]a > b[/tex], and subtracting b from both sides of the inequality, observe that [tex]a-b > 0[/tex].
Similarly, [tex]c > d[/tex], and subtracting d from both sides of the inequality, observe that [tex]c-d > 0[/tex].
From this, add "a-b" (a positive number, as proven above) to both sides of the inequality.
[tex](a-b)+(c-d) > (a-b)+0[/tex]
Addition by zero (the additive identity) doesn't change anything, so the right side remains "a-b"...
[tex](a-b)+(c-d) > a-b[/tex]
... and "a-b" is positive...
[tex](a-b)+(c-d) > a-b > 0[/tex]
... so, by the transitive property of inequality...
[tex](a-b)+(c-d) > 0[/tex]
Recall that subtraction is addition by a negative number...
[tex]a+(-b)+c+(-d) > 0[/tex]
...and that addition is associative and commutative, so things can be added in any order, so the middle two terms on the left side can be rearranged...
[tex]a+c+(-b)+(-d) > 0[/tex]
Adding b + d to both sides of the inequality
[tex](a+c+(-b)+(-d))+(b+d) > 0+(b+d)[/tex]
... and simplifying
[tex]a+c > b+d[/tex]
So, a+c is larger than b+d.
2. Which is larger, a+d or b+c?
Consider the following two examples:
Example 1
Suppose a=10; b=3; c=2; d=1.
Note that [tex]a > b > c > d[/tex] ([tex]10 > 3 > 2 > 1[/tex]) and, also observe that [tex]a+d=(10)+(1)=11[/tex], and [tex]b+c=(3)+(2)=5[/tex], so a+d is larger than b+c.
Example 2
However, suppose a=10; b=9; c=8; d=1.
Note that [tex]a > b > c > d[/tex] ([tex]10 > 9 > 8 > 1[/tex]) but that [tex]a+d=(10)+(1)=11[/tex], and [tex]b+c=(9)+(8)=17[/tex], 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.
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