Question of the week - 31 (How many non-positive integers satisfy...) GMAT Problem-Solving

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Question: How many non-positive integers satisfy the inequality (9–x²)(x+1)(x+2)²/ ((x+3)(x–2)) ≤ 0

A. 1
B. 2
C. 3
D. 4
E. Infinite

Answer: C

Solution and Explanation:

Approach Solution 1:
Apply the information in the question to the GMAT question at hand. These problems apply to numerous disciplines of mathematics. This question has to do with algebra. It is challenging to select the best option because of the way the options are presented. Candidates must be able to comprehend the appropriate approach to eliciting the desired response. Out of the five possible answers, there is only one that is correct.
Just examining the extreme X values for which the equation becomes zero will give you the solution.
Since we are only considering non-positive integers in this case, the values will be 3 and -3, but you can still see how this affects the whole equation if x is greater than -3.
Hence, change X to -4.
(9–(−4)²)(−4+1)(−4+2)²/ ((−4+3)(−4–2)) ≤ 0......(−7)∗(−3)∗4/ ((−1)∗(−6)) =14 > 0
Because the equation will be positive beyond x < -3, ignore all the values.
That leaves us with 0, 1, 2, and 3.
Throw out -3 since it makes the denominator 0 and leaves our equation undefined.
-1 and -2 will result in a value of 0, since the numerator has (x+1) and (x+2) in the numerator, respectively.
Hence, term becomes 9*1*2/(3*(-2))=-3 when x is 0.
Values will therefore be 0, 1, 2, and 3.
Correct option: C

Approach Solution 2:
Apply the information in the question to the GMAT question at hand. These problems apply to numerous disciplines of mathematics. This question has to do with fundamental math. It is challenging to select the best option because of the way the options are presented. Candidates must be able to comprehend the appropriate approach to eliciting the desired response. Out of the five possible answers, there is only one that is correct.
Given: (9–x²)(x+1)(x+2)² / ((x+3)(x–2)) ≤ 0
The quantity of non-positive integer x values that fulfill the specified inequality must be determined.
Working and Approach: • (9-x²)(x+1)(x+2)²/ ((x+3)(x–2)) ≤ 0
The denominator can't equal zero. Hence, x ≠ {-3, 2}
(3–x)(3+x)(x+1)(x+2)²/ ((x+3)(x+2)) < 0
The common term (x + 3) can be eliminated from the denominator and numerator by using the formula
(3-x)(x+1)(x+2)²/(x-2) <=0
Also, we are aware that (x+2)² is always greater 0. When x is equal to -2, it equals 0, hence the inequality will be 0. Thus, we have (3-x)(x+1)/(x-2) <= 0
The result is obtained by multiplying the numerator and denominator by (x - 2):
(3–x)(x+1)(x−2)/(x–2)² ≤ 0
o Denotes that (3 - x)(x + 1)(x - 2) = 0
o Multiplying both sides by -1 results in (x - 3)(x + 1)(x - 2) >= 0
x = -1, 2, 3, and -3 are the inequality's zero points.
Correct option: C

Approach Solution 3:
Apply the information in the question to the GMAT question at hand. These problems apply to numerous disciplines of mathematics. This question has to do with fundamental math. It is challenging to select the best option because of the way the options are presented. Candidates must be able to comprehend the appropriate approach to eliciting the desired response. Out of the five possible answers, there is only one that is correct.
Since the numerator is (3+X), X cannot be equal to either -3 or 2. (X-2).
Integers that are not positive range in value from 0 to negative infinity.
Hence, simply enter X= 0, X= -1, X=- 2, or X= -4 into the equation and check to see if the solution is non-positive.
The equation is negative when X = 0.
The equation is negative when X = -1.
The equation is negative for X = -2.
The equation is positive for X >= -4.
Correct option: C

“Question of the week - 31 (How many non-positive integers satisfy...)" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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