If ax + b = 0, is x > 0 (1) a + b > 0 (2) a - b > 0 GMAT Data Sufficiency

Question: If ax + b = 0, is x > 0?

1. a + b > 0
2. a – b > 0

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“If ax + b = 0, is x > 0?”– is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Answer:

Approach Solution 1:

Given: b = -ax

Question: is x > o

(1) a + b > 0 ----- a – ax > 0 ----- a(1 – x) > 0 ----- either a > 0 and 1 – x > 0, so x < 1 OR a < 0 and 1 – x < 0, so x > 1

NOT SUFFICIENT.

(2) a – b > 0 ---- a + ax > 0 ----- a(1 + x) > 0 ---- either a > 0 and 1 + x > 0, so x > -1 OR a < 0 and 1 + x < 0, so x < -1

NOT SUFFICIENT

(1) + (2) Sum (1) and (2) (we can do this as the signs of these inequalities are in the same direction)

(a + b) + (a – b) > 0 ---- a > 0, so we have the first range from (!): x < 1 and first case from (2): x > -1 ---- -1 < x < 1 so x may or may not be negative.

NOT SUFFICIENT

Correct option: E

Approach Solution 2:

You can look at this geometrically – because the equation is that of a line, the question is set up so that you can look at it as co-ordinate geometry question. The equation y = ax + b is the equation of a line with the slope a and y-intercept b. when we plug in y = 0, we are finding the x-intercept of the line. So the question is just asking if the x-intercept of the line is positive. In a diagram that would be true if our line crosses the x-axis to the right of the origin (0,0).

Now, using both the statements, if a > -b and a > b, then a > |b|. so a is clearly positive, and our line has a positive slope and is climbing as you move to the right. We can now just draw two different scenarios on the coordinate plane, one where b is positive and one where b is negative. If you choose, say, b = 1 and a = 5, then if you quickly sketch your line you’ll see that the x-intercept of y = ax + b is negative, and if b = -1 and a = 5 then if you quickly sketch your line you’ll see that the x-intercept of y = ax +b is positive, so NOT SUFFICIENT.

Correct option: E

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