• Magoosh
Study with Magoosh GMAT prep

Available with Beat the GMAT members only code

• 5 Day FREE Trial
Study Smarter, Not Harder

Available with Beat the GMAT members only code

• Free Trial & Practice Exam
BEAT THE GMAT EXCLUSIVE

Available with Beat the GMAT members only code

• Get 300+ Practice Questions

Available with Beat the GMAT members only code

• 5-Day Free Trial
5-day free, full-access trial TTP Quant

Available with Beat the GMAT members only code

• Free Practice Test & Review
How would you score if you took the GMAT

Available with Beat the GMAT members only code

• Award-winning private GMAT tutoring
Register now and save up to \$200

Available with Beat the GMAT members only code

• 1 Hour Free
BEAT THE GMAT EXCLUSIVE

Available with Beat the GMAT members only code

• Free Veritas GMAT Class
Experience Lesson 1 Live Free

Available with Beat the GMAT members only code

## Inequalities: Which of the following describes all values of

This topic has 2 expert replies and 11 member replies
II Master | Next Rank: 500 Posts
Joined
10 Dec 2007
Posted:
400 messages
19
Target GMAT Score:
700
GMAT Score:
680

#### Inequalities: Which of the following describes all values of

Mon Apr 21, 2008 5:39 am
Which of the following describes all values of x for which 1 - x² ≥ 0 ?

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

What is the best way to solve this question please ?
Thanks.
II

Last edited by II on Thu May 01, 2008 6:24 am; edited 1 time in total

II Master | Next Rank: 500 Posts
Joined
10 Dec 2007
Posted:
400 messages
19
Target GMAT Score:
700
GMAT Score:
680
Thu May 01, 2008 6:46 am
So we have 2 good approaches here (thanks to Stuart and Simplyjat)

1) Rewrite the original as: x^2 ≤ 1 and use a bit of logic.

For any number with magnitude greater than 1 (magnitude means we ignore the sign), the square of that number will also be greater than 1.
Both 1 and -1 squared = 1.
Fractions (positive and negative) squared are between 0 and 1.

So, for the statement to be true, |x| ≤ 1, which is just another way of saying -1 ≤ x ≤ 1: choose (e).

2) 1-x^2 ≥ 0

Rewrite 1-x^2 as (1+x)(1-x).
In order for (1+x)(1-x) to be ≥ 0, then x has an upper limit of 1, and a lower limit of -1 ... in other words it is between 1 and -1; -1 ≤ x ≤ 1 : choose E

resilient Legendary Member
Joined
06 May 2007
Posted:
789 messages
Followed by:
6 members
15
Target GMAT Score:
710
Mon Apr 21, 2008 9:19 pm
as we boil down to (x+1)(x-1)≤0 and x less than equal to 1 and -1. Why isnt b correct where it satisifies both the 1 and -1? I beleive I am getting the question wrong again. If we graph out x less than or equal to 1 and -1 we see that -1 is the statement where both are satisfied. I understand the math but mixed up on last step!

_________________
Appetite for 700 and I scraped my plate!

### GMAT/MBA Expert

Jeff@TargetTestPrep GMAT Instructor
Joined
09 Apr 2015
Posted:
647 messages
Followed by:
9 members
39
Thu Dec 07, 2017 7:24 am
II wrote:
Which of the following describes all values of x for which 1 - x^2 â‰¥ 0 ?

(A) x â‰¥ 1
(B) x â‰¤ -1
(C) 0 â‰¤ x â‰¤ 1
(D) x â‰¤ -1 or x â‰¥ 1
(E) -1 â‰¤ x â‰¤ 1
We are given that 1-x^2 >= 0 and need to determine an answer that describes all values of x. Letâ€™s isolate x in our inequality.

1-x^2 â‰¥ 0

1 â‰¥ x^2

Taking the square root of both sides of the inequality gives us:

1 â‰¥ x

x â‰¤ 1

OR

1 â‰¥ - x

-1 â‰¤ x

Thus, -1 â‰¤ x â‰¤ 1.

_________________
Jeffrey Miller Head of GMAT Instruction

The Iceman Master | Next Rank: 500 Posts
Joined
15 Oct 2012
Posted:
194 messages
Followed by:
6 members
47
Mon Oct 15, 2012 7:38 pm
pbrmoney wrote:
I can eliminate a, b, and d easily. But when it comes down to choosing between C and E, I'm a bit torn. C isn't actually wrong, is it? It just neglects to include -1 as answer choice E does... would that be fair to say.

Is the reason that E is correct just because it's a better answer (and incorporates -1), not necessarily because C is wrong?

Basically, plugging in gets me through AB and D. How do i decide between C and E?
Let's consider y=x^2 and y<=1

If you put a negative or a positive value of x of same magnitude the value of y remains the same. Such a function is also known as even function.

Hence, a situation describing x^2<=1 is analogous to |x|<=1.

This means we must have symmetry wrt values of x on both sides of '0'. Also the absolute value of x should not exceed 1.

Hence, -1<=x<=1

The problem with choice C is that it fails to take into account all the symmetrically placed negative values of x that I discussed above.

Note: Any even power of x makes the places the values of x symmetric to y axis.

pbrmoney Newbie | Next Rank: 10 Posts
Joined
14 Oct 2012
Posted:
5 messages
1
Mon Oct 15, 2012 5:06 pm
I can eliminate a, b, and d easily. But when it comes down to choosing between C and E, I'm a bit torn. C isn't actually wrong, is it? It just neglects to include -1 as answer choice E does... would that be fair to say.

Is the reason that E is correct just because it's a better answer (and incorporates -1), not necessarily because C is wrong?

Basically, plugging in gets me through AB and D. How do i decide between C and E?

akshatsingh Senior | Next Rank: 100 Posts
Joined
10 Apr 2008
Posted:
77 messages
4
Mon Apr 21, 2008 8:07 pm
Which of the following describes all values of x for which 1 - x² ≥ 0 ?

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

We can just change the sign an say:
x²-1≤0.
(x+1)(x-1)≤0
or x lies between -1 and 1. Plot on the number line to verify.

Cheers!
Aks

II Master | Next Rank: 500 Posts
Joined
10 Dec 2007
Posted:
400 messages
19
Target GMAT Score:
700
GMAT Score:
680
Mon Apr 21, 2008 1:06 pm
Thanks guys ... as ever Stuart breaks it down so well !
Good to have different approaches to solve questions ... when in the midst of the pressure of the GMAT ... its good to have secondary plans which you can draw upon to help you out.
And yes ... plugging in numbers here would help a lot.

netigen Legendary Member
Joined
18 Feb 2008
Posted:
631 messages
Followed by:
3 members
29
Mon Apr 21, 2008 10:38 am
As Stuart says:

If you are completely bowled by such a question, just use substitution which I believe will be fast enough to get this one right in less than 2 mins.

A. x =2 fails
B. x = -2 fails
C. x = 0 and x = 1 and x = 1/2 pass (lets keep it)
D. fails because A or B fails
E. x = -1 and C above pass so this is the right answer

simplyjat Master | Next Rank: 500 Posts
Joined
27 Dec 2007
Posted:
423 messages
Followed by:
2 members
36
Test Date:
May 20, 2008
GMAT Score:
770
Mon Apr 21, 2008 10:14 am
simplyjat wrote:
1 - x² ≥ 0 is equivalent to (1+x)(1-x) ≥ 0.
And that means either both (1+X) & (1-x) are negative or both (1+X) & (1-x) are negative...
That translates to x ≤ -1 or x ≥ 1 stated by option D
Sorry messed up the last part of translation ... E is the correct answer.

_________________
simplyjat

### GMAT/MBA Expert

Stuart Kovinsky GMAT Instructor
Joined
08 Jan 2008
Posted:
3225 messages
Followed by:
608 members
1710
GMAT Score:
800
Mon Apr 21, 2008 9:51 am
II wrote:
Which of the following describes all values of x for which 1 - x² ≥ 0 ?

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

What is the best way to solve this question please ?
Thanks.
II
Let's rewrite the original as:

x^2 ≤ 1

and let's use a bit of logic.

For any number with magnitude greater than 1 (magnitude means we ignore the sign), the square of that number will also be greater than 1.

Both 1 and -1 squared = 1.

Fractions (positive and negative) squared are between 0 and 1.

So, for the statement to be true, |x| ≤ 1, which is just another way of saying -1 ≤ x ≤ 1: choose (e).

Free GMAT Practice Test under Proctored Conditions! - Find a practice test near you or live and online in Kaplan's Classroom Anywhere environment. Register today!
mandy12 Junior | Next Rank: 30 Posts
Joined
31 Mar 2008
Posted:
24 messages
Mon Apr 21, 2008 9:41 am
IMO E

II Master | Next Rank: 500 Posts
Joined
10 Dec 2007
Posted:
400 messages
19
Target GMAT Score:
700
GMAT Score:
680
Mon Apr 21, 2008 9:11 am
If x is less than or equal to -1 ... according to answer D ... then x could have a value of -2.

If x = -2, then is 1 - x² ≥ 0 ? NO 1 - 4 = -3 ... which is NOT ≥ 0.

simplyjat Master | Next Rank: 500 Posts
Joined
27 Dec 2007
Posted:
423 messages
Followed by:
2 members
36
Test Date:
May 20, 2008
GMAT Score:
770
Mon Apr 21, 2008 8:33 am
1 - x² ≥ 0 is equivalent to (1+x)(1-x) ≥ 0.
And that means either both (1+X) & (1-x) are negative or both (1+X) & (1-x) are negative...
That translates to x ≤ -1 or x ≥ 1 stated by option D

_________________
simplyjat

### Best Conversation Starters

1 Roland2rule 181 topics
2 lheiannie07 110 topics
3 ardz24 60 topics
4 LUANDATO 55 topics
5 swerve 52 topics
See More Top Beat The GMAT Members...

### Most Active Experts

1 Brent@GMATPrepNow

GMAT Prep Now Teacher

153 posts
2 GMATGuruNY

The Princeton Review Teacher

125 posts
3 Scott@TargetTestPrep

Target Test Prep

123 posts
4 Rich.C@EMPOWERgma...

EMPOWERgmat

111 posts
5 EconomistGMATTutor

The Economist GMAT Tutor

83 posts
See More Top Beat The GMAT Experts