Algebra - GMAT Quantitative

Statement 1 is a consequence of Statement 2, so we need only show that Statement 2 provides insufficient information.

Let . Then

and

.

However, without knowing the sign of , we cannot determine whether or is the greater quantity.

is positive or zero, so if Statement 1 is assumed, and is negative, then , the product of a negative number and a nonnegative number, must be nonpositive. It can be negative or zero, however.

If Statement 2 is assumed, the expression can be of any sign, since, although must be positive, no information is given about the sign of .

If both statements are assumed, then since is nonzero, is positive; also, is negative. is the product of two numbers of unlike sign, and therefore, it can be determined that it is a negative number.

Similarly, .

Therefore, for , it must hold that

.

Statement 1 contradicts this; Statement 2 neither confirms nor contradict this.

Knowing one term of a sequence will not help you find any other terms, so neither statement alone will answer the question. But knowing two terms and knowing that the sequence is arithmetic will allow you to find the common difference .

The twentieth term is greater than the fourth term, so take the difference and divide by 16:

Now solve this inequality for to find the minimum number of terms needed to exceed 1,000:

, and . Therefore, since is a positive number, the sign of is the sign of . This makes Statement 1 neither necessary nor helpful. We need to know whether is greater than, equal to, or less than 0, or, equivalently, whether is greater than, equal to, or less than 10. Statement 2 does not tell us this either.

Statement (1) does not give us any information about g(x), so it is not sufficient.

Statement (2) alone gives us the relationship between the functions f and g but does not give us any information about f(x), so it is not sufficient.

Both statement together allow us to get an expression for both f and g:

With an expression for both functions we can estimate f(g(1)):

and

So the correct answer is C.

Assume Statement 1 alone. If is positive, then and ; since is the sum of three positive numbers, then , and is a false statement. Therefore, cannot be positive.

Assume Statement 2. If is positive, then so is , and the inequality can be rewritten as

Consequently,

,

a contradiction since is positive. Therefore, is not positive.

Statement 1 alone gives insufficient information.

Case 1:

, which is true.

Case 2:

, which is true.

But in the first case, is the greatest of the three. In the second, is the greatest.

Statement 2 gives insuffcient information, since no information is given about the sign of .

Assume both statements to be true. , and from Statement 1, ; by transitivity, . From Statement 2, . This makes the greatest of the three.

This is a quadratic formula problem. Use equation . For our problem, Plug these values into the equation, and simplify:

. Here, we simplified the radical by

To solve for three unknowns, we need three equations. Here we have three equations if we use both statements 1 and 2. We don't need to solve any further. Because this is a data sufficiency question, it doesn't matter what the actual values of x, y, and z are. The important fact is the we could find them if we wanted to.

To solve for three variables, you must have three equations. Statements 1 and 2 together only give two equations, so the statements together are not sufficient.

If we only look at statement 1, we might think the equation is not linear because of the term. But statement 2 tells us the is a constant. Then the equation is linear. We need both statements to answer this question.

When solving an equation with 2 variables, a second equation or the solution of 1 variable is necessary to solve.

Using statement 1 we can set up a series of equations and solve for both and .

Additionally, the information in statement 2 indicates that there is only one possible solution that satisfies the requirement that both are positive integers.

To solve with 2 unknowns, we must create a system of equations with at least 2 equations. Using statement 2 as a second equation we can easily get our answer. Solve statement 2 for x or y, and plug in for the corresponding variable in the equation given by the problem.

So, solving statement 2 for x, we get x=12+y. Replacing x in the equation from the problem, we get 4(12+y) + 7y=169. We can distribute the 4, and combine terms to find 48+11y=169. Subtract 48 from both sides, we get 11y=121. So y=11. Reusing either equation and plugging in our y value gives our x value. So x - 11=12, or x=23. This shows that x > y, and statement 1 is true. But even though it's true, it is completely unneccessary information. Therefore the answer is that we only need the information from statement 2, and statement 1 is not needed.

If the slopes of the lines are not equal, then the lines intersect at one solution; if they are equal, then they do not intersect, or the lines are the same line. Write each equation in slope-intercept form, :

The slopes of the lines are .

We need to know both and in order to determine their equality or inequality, and only if they are unequal can we answer the question.

Set and .

The slopes are unequal, so the lines intersect at one point; the system has exactly one solution.

If the slopes of the lines are not equal, then the lines intersect at one solution. If the slopes are equal, then there are two possibilties: either they do not intersect or they are the same line. Write each equation in slope-intercept form:

The slopes of these lines are .

If Statement 1 is true, then we can rewrite the first slope as , meaning that the lines have unequal slopes, and that there is only one solution. Statement 2 tells us the value of , which is irrelevant.

The equivalent question is "does have a real root?"

If you know only that is negative, you need to know whether is even or odd; negative numbers have real odd-numbered roots, but not real even-numbered roots.

If you know only that is even, you need to know whether is negative or nonnegative; negative numbers do not have real even-numbered roots, but nonnegative numbers do.

If you know both, however, then you know that the answer is no, since as stated before, negative numbers do not have real even-numbered roots.

Therefore, the answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.

In order to solve an equation set, one requires a number of equations equal to the number of variables. Therefore, either of the statements allow the problem to be solved.