GMATPrepNow wrote:

If x is an integer, what is the value of x?

(1) x² + 8x + 16 > 1

(2) x² – 8x + 9 < -6

Target question: What is the value of x? Statement 1: x² + 8x + 16 > 1 Subtract 1 from both sides to get: x² + 8x + 15 > 0

We might see right away that MANY different values of x will satisfy this inequality.

To begin,

any POSITIVE value of x will satisfy the inequality x² + 8x + 15 > 0

Since we cannot answer the

target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x² – 8x + 9 < -6 Add 6 to both sides to get: x² - 8x + 15 < 0

Factor to get: (x - 3)(x - 5) < 0

With quadratic inequalities like this, it's useful to first examine and solve the corresponding EQUATION

So, solve: (x - 3)(x - 5) = 0

We get: x = 3 or x = 5

These are our CRITICAL VALUES of x (i.e., x-values that satisfy the corresponding EQUATION)

Now let's examine what happens within the x-values LESS THAN and GREATER THAN these critical values.

There are 3 ranges to consider:

i) x is

less than 3

ii) x is

between 3 and 5

iii) x is

greater than 5

i) x is

less than 3

If x < 3, then (x - 3) is NEGATIVE and (x - 5) is NEGATIVE

So, (x - 3)(x - 5) = (NEGATIVE)(NEGATIVE) = POSITIVE

So, when x is

less than 3, (x - 3)(x - 5) > 0

We're looking for x-values that satisfy the inequality (x - 3)(x - 5) < 0, so we can conclude that x is NOT less than 3

ii) x is

between 3 and 5

If 3 < x < 5, then (x - 3) is POSTIVE and (x - 5) is NEGATIVE

So, (x + 3)(x + 5) = (POSITIVE)(NEGATIVE) = NEGATIVE

So, when x is

between 3 and 5, (x - 3)(x - 5) < 0

Perfect, we're looking for x-values that satisfy the inequality (x - 3)(x - 5) < 0!

So we can conclude that x IS

between 3 and 5

iii) x is

greater than 5

If x > 5, then (x - 3) is POSITIVE and (x - 5) is POSITIVE

So, (x - 3)(x - 5) = (POSITIVE)(POSITIVE) = POSITIVE

So, x is

greater than 5, (x - 3)(x - 5) > 0

We're looking for x-values that satisfy the inequality (x - 3)(x - 5) < 0, so we can conclude that x is NOT

greater than 5

We have seen that x IS

between 3 and 5

Since 4 is the ONLY integer between 3 and 5, we can be certain that

x = 4Since we can answer the

target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,

Brent

_________________

Brent Hanneson – Founder of gmatprepnow.com