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Bunuel
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If (x^2 + 2x - 9)/3 ≤ x + 1, then x could be represented by which of 03 Jan 2016, 12:38
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D
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80% (01:57) correct 20% (02:19) wrong based on 277 sessions

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If (x^2 + 2x - 9)/3 ≤ x + 1, then x could be represented by which of the following?

A. − 4 ≤ x ≤ − 3
B. − 4 ≤ x ≤ 3
C. − 3 ≤ x ≤ 3
D. − 3 ≤ x ≤ 4
E. 3 ≤ x ≤ 4
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D
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If (x^2 + 2x - 9)/3 ≤ x + 1, then x could be represented by which of 07 Jan 2016, 02:05
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Bunuel
If (x^2 + 2x - 9)/3 ≤ x + 1, then x could be represented by which of the following?

A. − 4 ≤ x ≤ − 3
B. − 4 ≤ x ≤ 3
C. − 3 ≤ x ≤ 3
D. − 3 ≤ x ≤ 4
E. 3 ≤ x ≤ 4
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This question tests your concepts of factorization and inequalities.

HINT:
Solving an inequality with a less than sign:
The value of the variable will be greater than the smaller value and smaller than the greater value.
i.e. It will lie between the extremes.

Solving an inequality with a greater than sign:
The value of the variable will be smaller than the smaller value and greater than the greater value.
i.e. It can take all the values except the values in the range.
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If (x^2 + 2x - 9)/3 ≤ x + 1, then x could be represented by which of 07 Jan 2016, 05:03
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IMO D is correct answer
solving through eqautions
x^2 +2x-9<= 3x+3
(x+3)(x-4)<=0
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If (x^2 + 2x - 9)/3 ≤ x + 1, then x could be represented by which of 07 Jan 2016, 09:27
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Bunuel
If (x^2 + 2x - 9)/3 ≤ x + 1, then x could be represented by which of the following?

A. − 4 ≤ x ≤ − 3
B. − 4 ≤ x ≤ 3
C. − 3 ≤ x ≤ 3
D. − 3 ≤ x ≤ 4
E. 3 ≤ x ≤ 4
Show more


Start by getting rid of the fraction by multiplying by 3

\(x^2+2x-9 \leq 3x+3\)

Gather all terms on the left hand side and factor it

\(x^2-x-12 = (x-4)(x+3) \leq 0\)

From here we need to decide what to do with the inequality. If it was an equation, then we would know that \(x=4\) and \(x=-3\). Those points will be limits of the range we want to consider. Right away we can see that the only answer choice with these limits is answer D. It would have been slightly trickier if some answer choices looked like \(x\leq-3\) and \(x\geq4\). That way we would have to worry about the direction of the inequality and @TeamGMATIFY's hint becomes super helpful.

If you like geometry and conic sections, you will recognize that \(f(x)=x^2-x-12\) is a parabola opening upward. The question asking for what values of x is the parabola \(\leq0\). Because the parabola is opening upwards, we know that the parabola \(f(x)\) will be negative in between the points where it crosses the x-axis. Those are the points we found by solving \(x^-x-12=0\). If you plotted it it would looks like this:



We can see that it crosses the x-axis at x=-3 and x=4, and is negative everywhere in between.
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If (x^2 + 2x - 9)/3 ≤ x + 1, then x could be represented by which of 04 Sep 2019, 05:26
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Bunuel
If (x^2 + 2x - 9)/3 ≤ x + 1, then x could be represented by which of the following?

A. − 4 ≤ x ≤ − 3
B. − 4 ≤ x ≤ 3
C. − 3 ≤ x ≤ 3
D. − 3 ≤ x ≤ 4
E. 3 ≤ x ≤ 4
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\((x^2 + 2x - 9)/3 ≤ x + 1…(x^2 + 2x - 9)≤3x+3…x^2 -x-12≤0…(x-4)(x+3)≤0…-3≤x≤4\):
since the inequality is "less than \(≤\)", then \(x\) must lie between the extremes \(-3≤x≤4\)
(if the inequality was "greater than \(≥\)", then \(x\) would lie outside the extremes, \((x-4)(x+3)≥0…[x≤-3,x≥4]\))

Answer (D)
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If (x^2 + 2x - 9)/3 ≤ x + 1, then x could be represented by which of 04 Sep 2019, 06:00
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Make use of the multiple choices.

If x=4 satisfies the inequalities, then options A,B, C can be eliminated. That's indeed the case (5≤5 --> Okay), thus A~C are all out.

If x=-3 satisfies the inequalities, then option E is out and D is the answer. That's truly the case (-2≤-2 --> Okay)

Answer is (D)
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If (x^2 + 2x - 9)/3 ≤ x + 1, then x could be represented by which of 10 Sep 2019, 14:50
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Bunuel
If (x^2 + 2x - 9)/3 ≤ x + 1, then x could be represented by which of the following?

A. − 4 ≤ x ≤ − 3
B. − 4 ≤ x ≤ 3
C. − 3 ≤ x ≤ 3
D. − 3 ≤ x ≤ 4
E. 3 ≤ x ≤ 4
Show more

How would the answer here change if we were left with:

(x+3)(x-4)<0

So not "less than or equal to 0" as we had in the question but "less than 0" as in my line above? Will the solution then be:

-3<x<4 (i.e. excluding 3 and 4?)
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If (x^2 + 2x - 9)/3 ≤ x + 1, then x could be represented by which of 28 Sep 2024, 21:59
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