M39-69

Question

If  x = -\sqrt {0.2} , then  x  satisfies which of the following inequalities?

A.  x < -1 
B.  -1 < x < -0.9 
C.  -0.9 < x < -0.5 
D.  -0.5 < x < -0.1 
E.  -0.1 < x < 0

Bunuel · Answer

Official Solution:

If  x = -\sqrt {0.2} , then  x  satisfies which of the following inequalities?

A.  x < -1 
B.  -1 < x < -0.9 
C.  -0.9 < x < -0.5 
D.  -0.5 < x < -0.1 
E.  -0.1 < x < 0

x = -\sqrt {0.2} =

= -\sqrt {\frac{1}{5}} =

= -\frac{1}{\sqrt {5}}

Such a large root from 5 would still be more than 1, however very close to it. Therefore:

-\frac{1}{\sqrt {5}}\approx

\approx -\frac{1}{1.001}

Finally,  -\frac{1}{1.001}  will be more than -1, however very close to it.

Answer: B

Vidds0214 · Answer

How have we derived that 200th root of 5 is greater than 1?

Bunuel · Answer

Here is a useful little trick: any positive integer root of a number greater than 1 will be more than 1 .

For example:  \sqrt {2} \gt 1 .

MBAToronto2024 · Answer

KarishmaB  I did the following: the x cannot be less than -1. That's why I selected A. Any advice on how to approach this kind of questions?

KarishmaB · Answer

Yes, we need to understand the number line for this. Let's convert it to exponents because those are easier to understand.

x = - ^{200}\sqrt{0.2}

-x = ^{200}\sqrt{0.2}

x^{200} = 0.2

x is negative. But let's evaluate its absolute value only for the time being. For now x just means absolute value of x. We will add a minus sign to it later.
When a number greater than 1 is multiplied by itself it becomes larger and larger so x cannot be more than 1.

When a number in the range 0 to 1 is multiplied with itself, it becomes smaller and smaller
(0.9)^2 = 0.81
(0.9)^3 = 0.729
(0.9)^4 = 0.6... (something)
(0.9)^5 = 0.5... 
Falling rapidly.

When x is raised to power 200, we get 0.2. So x must be between 0 and 1 but very very close to 1. You see how quickly 0.9 is dropping and will become 0.2 very soon? Hence x absolute value must be greater than 0.9 but must remain less than 1. Hence x will lie between -0.9 and -1. It will be something like -0.9999...

Answer (B)

I have discussed these relations on the number line in detail in my Exponents and Roots study module.

MAN11 · Answer

when we square 0.2 we get 0.04, square again -0.008 which is greater then -0.1 but less then 0... can you explain a bit more?

Bunuel · Answer

We are not squaring 0.2. We are squaring 200th root from 0.2. Also, squaring a number cannot give negative result.

robin36520 · Answer

I like the solution - it’s helpful.

robin36520 · Answer

I like the solution - it’s helpful.

Goldenfuture · Answer

Bunuel - But how do we know its 1.001 - it can be 1.2 or 1.5 and so ans c and D is also possible. We only know the root is more than 1 - how much more is where I didn't know and marked it wrong in exam

Bunuel · Answer

The 1.001 is not a fact you are supposed to “know.” It is just an approximation to communicate that the 200th root of 5 is  very  close to 1.

It cannot be 1.2 or 1.5, because numbers that large, raised to the 200th power, would be enormously bigger than 5. So the root is only  slightly  above 1, which is exactly the point the official solution is making. Just FYI, 1.2^20, so 1.2 in 20th power is already 38.someting, 1.2^200 is HUGE.

M39-69

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards