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

Hello,

You have \(1-x^2\geq{0}\).
Since LHS and RHS are non-negative,we can take square root on both sides and get

\(1\geq{\sqrt{x^2}}\)

Also, \(\sqrt{x^2}\)=|x| so we have \(|x|\leq{1}\)
So x is between \(-1\leq{x}\leq{1}\)

Also, you can do it as \(1-x^2\geq{0}\) or \((1-x)(1+x)\geq{0}\) (using \(a^2-b^2=(a-b)(a+b)\) )

We need to find in which region does the equation hold true...try values of x <-1, -1<x<1 and x>1 to see where the relationship holds true

You need to brush your basics on mod values. Check out below links

graphic-approach-to-problems-with-inequalities-68037.html
math-number-theory-88376.html
if-x-is-an-integer-what-is-the-value-of-x-1-x-2-4x-94661.html#p731476
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Which of the following describes all values of x for which 1–x^2 >= 0?

(A) x >= 1

Plugged in 2. 1–(2)^2 >= 0 -3>=0? NO. Wrong

(B) x <= –1

Plugged in -2. 1–(-2)^2 >= 0 -3>=0? NO. Wrong

(C) 0 <= x <= 1
Plugged in 0, 1, and 1/2. All of them work. But E is better because it describes all the values of x

(D) x <= –1 or x >= 1
A and B answer this. Wrong.

(E) –1 <= x <= 1

X is a positive or negative fraction or a zero.
x = -1/2 x = 1/2 x = 0
Plug all of them. They work.
Answer is E.

Another solution:
1–x^2 >= 0
1>= x^2
Therefore, -1=<x=<1

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.

Final Answer:

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In such inequalities, as long as one can factorize the expression into linear factors, the most methodical way to approach such questions is to use the wavy line approach.

You can refer to the following posts for a comprehensive treatment of the Wavy Line Approach:

http://gmatclub.com/forum/inequalities-trick-91482-80.html?sid=fde22066899cf98d261c2d987caa4509#p1465609

http://gmatclub.com/forum/wavy-line-method-application-complex-algebraic-inequalities-224319.html


Let’s apply this approach in the given question.

Given:

\(1–x^2 \geq{0}\)
\(x^2 – 1 \leq{0}\)

\((x + 1)*(x – 1) \leq{0}\) ………… (1)

Approach:

Apply the wavy line approach.

Mark the zero points on the number and draw the wavy line.
Identify the \(+ve\) and \(-ve\) regions of the curve.
Since we need the range of values of \(x\), for which the expression in (1) is less than or equal to zero, we consider the \(-ve\) region(s) along with the zero points.

Working Out:



The range of values of \(x\) for which the given inequality is satisfied is \(-1 \leq x \leq 1\)

Correct Answer: Option E
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