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# Which of the following describes all values of x for which

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Manager
Joined: 27 Mar 2008
Posts: 78
Which of the following describes all values of x for which [#permalink]

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16 Aug 2008, 11:58
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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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Location: New York
Re: PS: Inequality [#permalink]

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16 Aug 2008, 14:49
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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

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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Posts: 73
Re: PS: Inequality [#permalink]

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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

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?
Math Expert
Joined: 02 Sep 2009
Posts: 44352
Re: PS: Inequality [#permalink]

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20 Apr 2012, 04:08
2
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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

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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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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Joined: 02 Sep 2009
Posts: 44352
Re: Which of the following describes all values of x for which [#permalink]

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17 Jun 2013, 05:51
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Re: Which of the following describes all values of x for which [#permalink]

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07 Jul 2016, 10:00
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Expert's post
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

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

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14 Feb 2018, 16:44
Hi All,

With inequality-based questions, sometimes the easiest approach is just to come up with a few examples that 'fit' the given prompt and then use those examples to eliminate answer choices.

Here, we're told that 1 - X^2 >= 0. We're asked for ALL of the possible values that fit this inequality.

The 'easiest' value that most Test Takers would immediately 'see' is 1 (since 1 - 1^2 = 0), so X COULD be 1.

Next, since we're dealing with a squared term, -1 would also be a solution (since 1 - [-1]^2 = 0).

So we immediately have at least two solutions: 1 and -1. We can eliminate Answers A, B and C.

For the last step, we have to determine what OTHER solutions are possible. You can either prove that fractions fit (try using X = 1/2) or proving that larger integers do NOT fit (try using X = 2). Either way, you can eliminate the final incorrect answer.

[Reveal] Spoiler:
E

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Re: Which of the following describes all values of x for which   [#permalink] 14 Feb 2018, 16:44
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