BTGModeratorVI wrote: ↑

Wed Apr 22, 2020 11:13 am

If ax + b = 0, is x > 0

(1) a + b > 0
(2) a - b > 0

Answer: E

Given that ax + b = 0, we have x > –b/a

To ascertain whether x > 0, a and b must be of opposite signs.

Question rephrased: Do a and b have opposite signs?

Let's take each statement one by one.

(1) a + b > 0

Case 1: Say a = 2; and b = 1. The answer is no.
Case 2: Say a = 2; and b = –1. The answer is yes. No unique answer. Insufficient.

(2) a – b > 0

Both cases discussed above are applicable here, too. Insufficient.

So. even both statements together can't give the finite answer. Insufficient.

The correct answer: E

Hope this helps!

-Jay
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BTGModeratorVI wrote: ↑

Wed Apr 22, 2020 11:13 am

If ax + b = 0, is x > 0

(1) a + b > 0
(2) a - b > 0

Answer: E

Target question: Is x > 0

Given: ax + b = 0

Statement 1: a + b > 0
At this point, we have 1 equation, 1 inequality, and THREE variables.
Even if we had 2 EQUATIONS and 3 variables, we probably wouldn't be able to make any conclusions about whether x is positive or negative.
Given this, let's TEST some values
There are several values of a, b and x that satisfy statement 1 (and the given equation ax + b = 0). Here are two:
Case a: a = 2, b = -1 and x = 0.5. In this case, the answer to the target question is YES, x is positive
Case b: a = 2, b = 1 and x = -0.5. In this case, the answer to the target question is NO, x is NOT positive
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: a - b > 0
This is the same scenario as statement 1, so let's TEST some values
There are several values of a, b and x that satisfy statement 1 (and the given equation ax + b = 0). Here are two:
Case a: a = 2, b = -1 and x = 0.5. In this case, the answer to the target question is YES, x is positive
Case b: a = 2, b = 1 and x = -0.5. In this case, the answer to the target question is NO, x is NOT positive
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I was able to use the same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED.
In other words,
Case a: a = 2, b = -1 and x = 0.5. In this case, the answer to the target question is YES, x is positive
Case b: a = 2, b = 1 and x = -0.5. In this case, the answer to the target question is NO, x is NOT positive

Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

BTGModeratorVI wrote: ↑

Wed Apr 22, 2020 11:13 am

If ax + b = 0, is x > 0

(1) a + b > 0
(2) a - b > 0

Answer: E

Solution:

Assuming that a is non-zero, we can write x = -b/a. This expression tells us that x is -1 times the ratio of b to a; i.e. if b and a have the same sign, then x is negative, and if b and a have opposite signs, then x is positive. We only need to look for examples where a and b have the same and opposite signs.

Statement One Alone:

For an example where a and b have the same sign, we can simply let a = b = 1. In this case, x + 1 = 0, and hence, x = -1 i.e. x > 0 is not true.

For an example where a and b have opposite signs, we can let a = 2 and b = -1. In this case, 2x - 1 = 0, and hence, x = 1/2 i.e. x > 0 is true.

Statement one alone is not sufficient.

Statement Two Alone:

For an example where a and b have the same sign, we can let a = 2 and b = 1. In this case, 2x + 1 = 0, and hence, x = -1/2 i.e. x > 0 is not true.

For an example where a and b have opposite signs, we can let a = 1 and b = -1. In this case, x - 1 = 0, and hence, x = 1 i.e. x > 0 is true.

Statement two alone is not sufficient.

Statements One and Two Together:

Notice that if a + b > 0 and a - b > 0 are true at the same time, then adding the inequalities side by side, we determine that 2a > 0; i.e. a > 0.

If a = 2 and b = 1, then 2x + 1 = 0 and hence, x = -1/2. Notice that a = 2 and b = 1 satisfy both a + b > 0 and a - b > 0. In this case, x < 0.

If a = 2 and b = -1, then 2x - 1 = 0 and hence, x = 1/2. Notice that a = 2 and b = - 1 satisfy both a + b > 0 and a - b > 0. In this case, x > 0.

Statements one and two together are not sufficient.

Answer: E