ykaiim wrote:
x is an integer and x raised to any odd integer is greater than zero; is w - z greater than 5 times the quantity 7^(x-1)-5^x?
(1) z < 25 and w = 7^x
(2) x = 4
Given: x is an integer and x raised to any odd integer is greater than zero KEY CONCEPT: A number raised to an ODD power retains its sign.
That is, POSITIVE^ODD = POSITIVE and NEGATIVE^ODD = NEGATIVE
The given info tells us that x^ ODD = POSITIVE
So, it must be the case that
x is a POSITIVE integerTarget question: Is w - z > 5[7^(x-1) - 5^x?]Let's tidy the target question by expanding the right side of the inequality to get....
REPHRASED target question: Is w - z > 5[7^(x-1)] - 5^(x+1)? Statement 1: z < 25 and w = 7^xTake the REPHRASED target question and replace w with 7^x to get:
Is 7^x - z > 5[7^(x-1)] - 5^(x+1)?IMPORTANT: Notice that, on the left side of the inequality, we're subtracting z
We're told that z < 25
So, we can help MINIMIZE the value of the left side by making z as big as possible
So, let's make z = 25 (aside: we could make z equal something really close to 25, like 24.99999999999, but let's make things easy on ourselves and make z EQUAL 25.
We get:
Is 7^x - 25 > 5[7^(x-1)] - 5^(x+1)?Now let's get like terms on the same side of the inequality.
First subtract 5[7^(x-1)] from both sides to get:
Is 7^x - 5[7^(x-1)] - 25 > -5^(x+1)?Then add 25 to both sides to get:
Is 7^x - 5[7^(x-1)] > 25 - 5^(x+1)?Factor 7^(x-1) from left side to get:
Is 7^(x-1)[7 - 5] > 25 - 5^(x+1)?Simplify left side to get:
Is 7^(x-1)[2] > 25 - 5^(x+1)?Now recognize that 7^(x-1) will be POSITIVE for all values of x.
So, 7^(x-1)[2] must be POSITIVE
Now recognize that since
x is a POSITIVE integer, 5^(x+1) must be greater than or equal to 25 (since x = 1, is the smallest possible value of x)
So, 25 - 5^(x+1) must be less than or equal to zero.
So, our question becomes
Is some POSITIVE number > some number that's less than or equal to zero?The answer to this
REPHRASED target question is a definitive YES
Since we can answer the
REPHRASED target question with certainty, statement 1 is SUFFICIENT
Statement 2: x = 4 Since we have no information about z or w, we cannot answer the
REPHRASED target question with certainty.
So, statement 2 is NOT SUFFICIENT
Answer: A
Cheers,
Brent
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