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Which of the following describes all values of x for which

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Which of the following describes all values of x for which [#permalink]

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Which of the following describes all values of x for which 1-x^2 ≥ 0 ?

A. x ≥ 1
B. x ≤ -1
C. 0 ≤ x ≤ 1
D. x ≤ -1 or x ≥ 1
E. -1 ≤ x ≤ 1
[Reveal] Spoiler: OA

Last edited by Bunuel on 20 Apr 2012, 03:41, edited 2 times in total.
Edited the question and added the OA
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Re: PS: Inequality [#permalink]

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New post 16 Aug 2008, 14:49
droopy57 wrote:
Which of the following describes all values of x for which 1-x^2 >= 0 ?

(a) x ≥ 1
(b) x ≤ -1
(c) 0 ≤ x ≤ 1
(d) x ≤ -1 or x ≥ 1
(e) -1 ≤ x ≤ 1

Please expand on answers


E.

1-x^2 >= 0 ---> x^2-1<=0
--> (x+1)(x-1)<=0
Above equation true for
i) x+1<=0 and x-1>=0 ---> x<= -1 and x>=1 ---> this is not possible ---Strike out this solution
ii) x+1>=0 and x-1<=0 ---> x>=-1 and x<=1 --> -1<=x<=1
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Re: PS: Inequality [#permalink]

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New post 20 Apr 2012, 02:18
x2suresh wrote:
droopy57 wrote:
Which of the following describes all values of x for which 1-x^2 >= 0 ?

(a) x ≥ 1
(b) x ≤ -1
(c) 0 ≤ x ≤ 1
(d) x ≤ -1 or x ≥ 1
(e) -1 ≤ x ≤ 1

Please expand on answers


E.

1-x^2 >= 0 ---> x^2-1<=0
--> (x+1)(x-1)<=0
Above equation true for
i) x+1<=0 and x-1>=0 ---> x<= -1 and x>=1 ---> this is not possible ---Strike out this solution
ii) x+1>=0 and x-1<=0 ---> x>=-1 and x<=1 --> -1<=x<=1


Can someone please explain the signs in red above? this is not absolute value, why do we need to test these?
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Re: PS: Inequality [#permalink]

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New post 20 Apr 2012, 04:08
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catty2004 wrote:
x2suresh wrote:
droopy57 wrote:
Which of the following describes all values of x for which 1-x^2 >= 0 ?

(a) x ≥ 1
(b) x ≤ -1
(c) 0 ≤ x ≤ 1
(d) x ≤ -1 or x ≥ 1
(e) -1 ≤ x ≤ 1

Please expand on answers


E.

1-x^2 >= 0 ---> x^2-1<=0
--> (x+1)(x-1)<=0
Above equation true for
i) x+1<=0 and x-1>=0 ---> x<= -1 and x>=1 ---> this is not possible ---Strike out this solution
ii) x+1>=0 and x-1<=0 ---> x>=-1 and x<=1 --> -1<=x<=1


Can someone please explain the signs in red above? this is not absolute value, why do we need to test these?


Actually you can transform it to an absolute value problem: \(1-x^2\geq{0}\) --> \(x^2\leq{1}\), since both parts of the inequality are non-negative then we can take square root: \(|x|\leq{1}\) --> \(-1\leq{x}\leq{1}\).

Now, other approach would be: \(1-x^2\geq{0}\) --> \(x^2-1\leq{0}\) --> \((x+1)(x-1)\leq{0}\) --> the roots are -1 and 1 --> "<" sign indicates that the solution lies between the roots, so \(-1\leq{x}\leq{1}\).


Solving inequalities:
x2-4x-94661.html#p731476 (check this one first)
inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html?hilit=extreme#p873535
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

Now, about x2suresh's approach: we have \((x+1)(x-1)\leq{0}\), so the product of two multiples is less than (or equal to) zero, which means that the multiples must have opposite signs. Then x2suresh checks the case A. when the first multiple (x+1) is negative and the second (x-1) is positive and the case B. when the first multiple (x+1) is positive and the second (x-1) is negative to get the range for which \((x+1)(x-1)\leq{0}\) holds true. Notice that, for this particular problem, we don't realy need to test case A, since it's not possible (x+1), the larger number, to be negative and (x-1), the smaller number to be positive. As for case B, it gives: \(x+1\geq{0}\) and \(x-1\leq{0}\) --> \(x1\geq{-1}\) and \(x\leq{1}\) --> \(-1\leq{x}\leq{1}\).

Hope it helps.
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Re: PS: Inequality [#permalink]

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New post 20 Apr 2012, 09:55
Thank you soooooooooooo much Bunuel!!

Bunuel wrote:
catty2004 wrote:
Can someone please explain the signs in red above? this is not absolute value, why do we need to test these?


Actually you can transform it to an absolute value problem: \(1-x^2\geq{0}\) --> \(x^2\leq{1}\), since both parts of the inequality are non-negative then we can take square root: \(|x|\leq{1}\) --> \(-1\leq{x}\leq{1}\).

Now, other approach would be: \(1-x^2\geq{0}\) --> \(x^2-1\leq{0}\) --> \((x+1)(x-1)\leq{0}\) --> the roots are -1 and 1 --> "<" sign indicates that the solution lies between the roots, so \(-1\leq{x}\leq{1}\).


Solving inequalities:
x2-4x-94661.html#p731476 (check this one first)
inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html?hilit=extreme#p873535
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

Now, about x2suresh's approach: we have \((x+1)(x-1)\leq{0}\), so the product of two multiples is less than (or equal to) zero, which means that the multiples must have opposite signs. Then x2suresh checks the case A. when the first multiple (x+1) is negative and the second (x-1) is positive and the case B. when the first multiple (x+1) is positive and the second (x-1) is negative to get the range for which \((x+1)(x-1)\leq{0}\) holds true. Notice that, for this particular problem, we don't realy need to test case A, since it's not possible (x+1), the larger number, to be negative and (x-1), the smaller number to be positive. As for case B, it gives: \(x+1\geq{0}\) and \(x-1\leq{0}\) --> \(x1\geq{-1}\) and \(x\leq{1}\) --> \(-1\leq{x}\leq{1}\).

Hope it helps.
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Re: Which of the following describes all values of x for which [#permalink]

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New post 17 Jun 2013, 05:51
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: Which of the following describes all values of x for which [#permalink]

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Re: Which of the following describes all values of x for which [#permalink]

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New post 28 Jun 2016, 17:52
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: Which of the following describes all values of x for which [#permalink]

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New post 07 Jul 2016, 10:00
droopy57 wrote:
Which of the following describes all values of x for which 1-x^2 ≥ 0 ?

A. x ≥ 1
B. x ≤ -1
C. 0 ≤ x ≤ 1
D. x ≤ -1 or x ≥ 1
E. -1 ≤ x ≤ 1


To solve, we first isolate the x^2 in the inequality 1 – x^2 ≥ 0. So we have:

1 ≥ x^2

Next, we take the square root of both sides, to isolate x.

√1 ≥ √x^2

This gives us:

1 ≥ |x|

Because the variable x is inside the absolute value sign, we must consider that x can be either positive or negative. Therefore, we’ll need to solve the inequality twice.

When x is positive:

1 ≥ |x| means

1 ≥ x

This can be re-expressed as x ≤ 1.

When x is negative:

1 ≥ |x| means

1 ≥ -x (Divide both sides by -1 and switch the inequality sign)

-1 ≤ x

We combine the two resulting inequalities to get:

-1 ≤ x ≤ 1

Answer is E.
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Re: Which of the following describes all values of x for which   [#permalink] 07 Jul 2016, 10:00
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