Which of the following describes all values of x such that

Question

Which of the following describes all values of x such that  \frac{x + 2}{x - 3} < -4 ?

A. -3 < x < 2
B. 2 < x < 3
C. -2 < x < 3
D. -3 < x < -2
E. x > 2

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Bunuel · Answer

\frac{x + 2}{x - 3} < -4 \frac{x + 2}{x - 3} + 4 <0 \frac{x + 2 + 4(x - 3)}{x - 3}<0 \frac{5x - 10}{x - 3}<0 \frac{x - 2}{x - 3}<0 The transition points are x = 2 and x = 3 (transition points are the values of x for which the numerator or denominator equals 0). This gives us three ranges: x < 2 2 < x < 3 x > 3 Next, test an extreme value for x: if x is a large enough number, say 10, then both the numerator and the denominator will be positive, resulting in a positive value for the whole expression. Therefore, when x > 3, the expression is positive. Now, here’s the trick: since the expression is positive in the 3rd range, it will be negative in the 2nd range and positive again in the 1st range, following the pattern: +, -, +. Thus, the expression is negative for 2 < x < 3 . Answer: B. Check the highlighted links below for the trick described above: 9. Inequalities Theory Inequalities Made Easy! Inequalities: Tips and hints Arithmetic with Inequalities Wavy Line Method Application - Complex Algebraic Inequalities Solving Quadratic Inequalities: Graphic Approach Inequalities - Quadratic Inequalities Graphic approach to problems with inequalities Inequalities trick INEQUATIONS(Inequalities) Part 1 INEQUATIONS(Inequalities) Part 2 Questions Inequality and absolute value questions from my collection DS Questions PS Questions For more check Ultimate GMAT Quantitative Megathread Hope it helps.

nandini14 · Answer

Why can't I write it as = x+2<-4x+12 , ie, x<2?

Bunuel · Answer

We can’t do that because the sign of (x - 3) is unknown. If (x - 3) is positive, your approach works and gives what you wrote. But if (x - 3) is negative, then multiplying by a negative flips the inequality sign, leading to a different result. Please check the correct solution above and use the linked resources for more on handling inequalities.

Which of the following describes all values of x such that

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Which of the following describes all values of x such that