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DIn this question, you are asked to compare 2x with 3x+1, given that x is a negative integer. One way to approach this problem i
DIn this question, you are asked to compare 2x with 3x+1, given that x is a negative integer. One way to approach this problem i
admin
2014-08-13
29
问题
选项
答案
D
解析
In this question, you are asked to compare 2
x
with 3
x+1
, given that x is a negative integer. One way to approach this problem is to plug a value of x in both expressions and compare the results.
You are given that x is a negative integer, so the greatest integer you can plug in for x is -1.
For x = -l, it follows that 2
x
= 2
-1
= 1/2 and 3
x+1
=3
-1+1
=3
0
= 1.
In this case, 2
x
is less than 3
x+1
. However, to conclude that Quantity B is greater, it is not sufficient for 2
x
to be less than 3
x+1
for one particular value of x; the relationship would need to be true for
all
negative integer values of x. To analyze this relationship further, plug in another value of x, for example, -2.
For x = -2, it follows that 2
x
= 2
-2
=
= 1/4 and
.
Again, 2
x
is less than 3
x+1
but note that these values are closer together than the previous values of 2
x
and 3
x+1
. It appears that the relationship between the quantities may differ for smaller values of x, so now try plugging in —3 for x.
For x = -3, it follows that 2
x
= 2
-3
=
=1/8 and 3
x+1
= 3
-3+1
= 3
-2
=
= 1/9.
In this case, 2
x
is greater than 3
x+1
.
Since 2
x
is less than 3
x+1
for x = -1 and 2
x
is greater than 3
x+1
for x = — 3, the relationship between these two quantities cannot be determined from the information given. The correct answer is Choice D.
Since both quantities are algebraic expressions, another way to approach the comparison is to set up a placeholder relationship, denoted by
, between the two quantities and then to simplify to see what conclusions you can draw. As you simplify and draw conclusions, keep in mind that x is a negative integer.
For any value of x(including negative integer values of x), the value of 3
x
is positive, so dividing by 3
x
does not affect any inequality that might be represented by the placeholder. Since each step in this simplification is reversible, the simplification reduces the problem to comparing
with 3, given that x is a negative integer. Note that
, where n = —x; so the problem can be reduced further to comparing
with 3, given that n is a positive integer.
Because 3/2 is greater than 1, the value of
becomes greater as n becomes larger. For small values of n,
is less than 3, but for large values of n,
is greater than 3. Therefore, the relationship between Quantity A and Quantity B cannot be determined from the given information, and the correct answer is Choice D.
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