dave13 wrote:
Bunuel wrote:
If \(|x| < x^2\), which of the following must be true?
I. \(x^2 > 1\)
II. \(x > 0\)
III. \(x < -1\)
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only
Given: \(|x|<x^2\);
Reduce by \(|x|\): \(1<|x|\) (side note: we can safely do this as absolute value is non-negative and in this case we know it's not zero too, as if x would be zero inequality wouldn't hold true, so \(|x|>0\));
So we have that \(x<-1\) or \(x>1\).
I. \(x^2>1\) --> always true;
II. \(x>0\) --> may or may not be true;
III. \(x<-1\) --> may or may not be true.
Answer: A (I only).
hello
generis,
your previous posts were excellent
many thanks
i am now trying to better develop understanding of absolute values
my reasoning of the above problem is based on the two properties below:
\(|x|\) = \(\sqrt{x^2}\) this property means that negative values rurn into positive one
?
\(|x|≥0\) this property absolutue value is always positive or equals zero, right?
For example if x is -5
\(|-5|\) = \(\sqrt{(-5)^2}\) = \(\sqrt{(-5)*(-5)}\) = \(\sqrt{5*5}\) = \(\sqrt{(5)^2}\) = 5
so \(|x|\) means distance between x and -x
QUESTION ASKS
If \(|x| < x^2\), which of the following must be true? SO I HAVE THREE OPTIONS COMBINED WITH THREE CONFUSIONS
I. \(x^2 > 1\)
II. \(x > 0\)
III. \(x < -1\)
To tackle answer choices at first from here \(|x| < x^2\) i need to set condition
Can I square both sides ? \(|x| < x^2\) ---> like this \((|x|)^2 < (x^2)^2\) --- and then i get \(x < x^4\) what for i am doing this have no idea ... any idea ?
looking at answer choices i see that we need to compare -1 VS +1 ? so we cant plug in any values ?
or any numbers we can plug in ?
FIRST OPTION I. \(x^2 > 1\)
let x be 1.
based on this formula |x|≥0
\(1^2 > 1\) vs \(-1^2 > 1\) so how can X^2 be aways greater than 1 ? 1*1 = techinally is 1
if x were -5 then \(-5^2 > 1\) --> \(25 > 1\)
SECOND OPTION II. \(x > 0\)
again based on this formula
\(|x|≥0\) , x can be more than or EQUAL ZERO
THIRD OPTION III. \(x < -1\)
i somehow can`t wrap my mind around this option even can`t ask a right question to understand it
if we compare x = -1 and x = +1 than either of the choises could be correct, same applies to other values (-5 or +5 )
SECOND OPTION II. \(x > 0\)
again based on this formula \(|x|≥0\) , x can be more than or EQUAL ZERO
many thanks and have an absolutely positive weekend
dave13 - uhh . . . I am not too sure about "mind-blowing" (you use vernacular well!), but here is your answer.
Sorry for the wait. I'm going to reverse the order of your questions, in a way.
The main concept here keeps getting missed, and as a result, many people are doing way too much work!
Main concept: the given fact and its conditions determine whether the statements in the options MUST be true.
If an option is true for one number but not for another, according to the conditions we find, it is PARTLY FALSE and gets eliminated.
We have a
given fact, a posited truth, so we have to accept it. AND... it has conditions.
The statement is true as long as certain conditions are met.
Here is the given fact:
\(|x| < x^2\)The fact and its conditions precede and determine the truth value of all the options.
Quote:
To tackle answer choices at first from here \(|x| < x^2\) i need to set condition
Excellent. I might add that we need to FIND the conditions (because the conditions follow from the fact).
You started by squaring both sides and trying an absolute value approach.
Then you assessed the options in a few different ways. I want to streamline this process.
Start with the fact.
Find the conditions.
Assess the answers: given the fact and its conditions, which options MUST be true?
Squaring both sides IS possible, but not ideal, IMO. Squaring requires us to consider the direction of signs
at critical points, and is neither as straightforward nor as efficient as the method
Bunuel used.
FACT AND CONDITIONS
\(|x| < x^2\)We need more information to assess options.
Typically with absolute value questions and/or inequalities, there are conditions.
We can use logic, algebra, or numbers to try simplifying the given fact.
Bunuel used algebra immediately.
We are asking:
(1) What are the conditions under which this fact is true?
If we test values in an absolute value inequality, we might well find the conditions.
Almost always, we can simplify with algebra, too.
Test numbers: under what conditions is this statement NOT true? TRUE?
Test a set of "weird" values (sometimes they do not behave like most positive and negative numbers).
Test: 0, 1, -1, \(\frac{1}{2}\), and \(-\frac{1}{2}\)
Is the statement true if:
\(x=0\)? NO. The statement is not true. \(|0|=0\) and \(0^2=0\).
So \(|0|\) is not LESS THAN \(0^2\) - they are equal
\(x=1\)? NO. \(|1|=1^2\). LHS is NOT less than RHS (they are equal).
(-1) is exactly the same as 1.
\(x=\frac{1}{2}\)? No. \(|\frac{1}{2}|=\frac{1}{2}\) and
\(\frac{1}{2})^2=\frac{1}{4}\). LHS is GREATER than RHS. The same thing will happen with
\(-\frac{1}{2}\)Now test 2 and -2. See what happens. Is the statement true?
From those tests, you should be able to sketch a number line, darken it to reflect conditions,
and extrapolate mathematical statements from it to indicate those conditions. (Hint: we know -1, -1/2, 0, 1/2, and 1 do not work.)
Algebra: more difficult, but faster. We need values (and their limits) to assess options
Background material
• absolute value is always non-negative, i.e. 0 or positive
• for this |x|, \(x\neq0\) because |0| is
equal to \(0^2\), not less than \(0^2\)
• Because \(x\neq0\), we are guaranteed that \(|x|\) is positive. Hence we can divide both sides of the inequality by |x|
• \(x^2=|x|^2=(|x|*|x|)\)
Use numbers to see it. Positive \(x=5\) is easy.
\(x = -5\): Now \(x^2=(|x|*|x|)=(5*5)=25=(-5)^2\)
\(|x| < x^2\)\(|x| < |x|*|x|\)
Divide both sides by |x|
\(\frac{|x|}{|x|}<\frac{|x|*|x|}{|x|}\)
\(1<|x|\), which is equivalent to
\(|x|>1\)
(2) Once simplified, find values for \(|x|>1\)
\(|x|>1\)
If \(|x| > 1\) , then
Case 1: \(x > 1\)
Case 2: \(-x > 1\)
Divide by (-1), flip the sign
\(x < -1\)
\(x>1\) OR \(x<-1\)If we draw the range of solutions on the number line, we have
Attachment:
number line 2018 06 29.jpg [ 28.32 KiB | Viewed 936 times ]
\(x\) can be any value in the two ranges on the number line(3) Assess options. \(|x| < x^2\). And \(x>1\) or \(x<-1\).
Which options MUST BE TRUE?
I. \(x^2 > 1\) MUST be true
We can look at the number line: ALWAYS true.
Bunuel wrote:
Quote:
Option I says that the square of that number is greater than 1. Well, square of ANY number less than -1 or more than 1 is greater than 1. So, no matter what x actually is, this statement is always true.
Pick any point on the blue part of the line as a value for \(x\). Square it. It will always be greater than 1.
Or (algebra that involves critical points): \(x^2>1\)
\(x<-1\) or \(x>1\)
If that solution set does not make sense,
please see this post and read all the linked material. Looking at the number line is easier
II. \(x>0\) - NOT ALWAYS TRUE.
Our conditions are \(x>1\) or \(x<-1\)
This statement is not true if \(x=-7\) (-7 is not greater than 0)
III. \(x<-1\) - NOT ALWAYS TRUE. If \(x=8\), it is not true. (8 is NOT < -1)
Answer AI hope that helps!
Quote:
\(|x|\) = \(\sqrt{x^2}\) this property means that negative values rurn into positive one
?
Not quite. That equation means: this is the positive root of \(x^2\)
\(-\sqrt{x^2} = -|x|\)
That equation means: this is the negative root of \(\sqrt{x^2}\)
Quote:
\(|x|≥0\) this property absolutue value is always positive or equals zero, right?
Yes.
In this case, \(x\) does NOT equal 0, so \(x\) is positiveQuote:
For example if x is -5
\(|-5|\) = \(\sqrt{(-5)^2}\) = \(\sqrt{(-5)*(-5)}\) = \(\sqrt{5*5}\) \(= 25 = (-5)^2\)
\(\sqrt{(5)^2}\) = 5 OMIT THIS PART
Mostly true. I have amended slightly. Also note that if \(x = -5, |x| = -x = -(-5)\)
Quote:
so \(|x|\) means distance between x and -x
No. |x| means the distance of \(x\) OR \(-x\) from \(0\), but
not the distance between them
Quote:
Can I square both sides ? \(|x| < x^2\) ---> like this \((|x|)^2 < (x^2)^2\) --- and then i get \(x < x^4\) what for i am doing this have no idea ... any idea ?
At this point I am not sure I have any idea!
If you are asking, "is it possible"?
Yes. You wouldn't want to, IMO.
We found the conditions without squaring. We just divided by \(|x|\), got to \(|x| > 1\), and found our conditions.
Why are you doing it?
Best guess: Because you have seen others do it in different contexts.
Squaring both sides
will work.
VeritasPrepKarishma discusses such squaring in this case
in this post.GMATinsight also discusses squaring both sides
here.I square both sides only when absolutely necessary.
Bunuel posted links you might need to read
in this post.Quote:
looking at answer choices i see that we need to compare -1 VS +1 ?
I don't understand here . . . We need to see whether the options
fulfill the conditions laid out in the diagram of the number line: \(x>1\) or \(x<-1\)
Maybe that is what you mean.
Quote:
so we cant plug in any values ?
or any numbers we can plug in ?
Sure you can! While searching for conditions or some simplification, we tested some unique numbers (-1, 1, 0 . . .).
From there, with some logic, we could have drawn the number line.
Or, if we have the conditions and we are assessing options, plug in numbers that are given or possible from the option.
See whether the numbers in the context of that option satisfy our conditions.
In fact, with numbers you can prove that Option I must be true, and that Options 2 and 3 might not be true. If needed, sketch a number line.
Quote:
FIRST OPTION I. \(x^2 > 1\)
let x be 1.
based on this formula |x|≥0
You are analyzing based on a rule that is important but . . .that rule is not the same as the conditions.
See above where we tested numbers. We KNOW we cannot use 1 or -1
x is greater than 1 and less than -1
You are testing
and violating explicit conditions that say you may not use those numbers.
Quote:
\(1^2 > 1\) vs \(-1^2 > 1\) so how can X^2 be aways greater than 1 ? 1*1 = techinally is 1
if x were -5 then \(-5^2 > 1\) --> \(25 > 1\)
AHA! So . . . -1 and 1 do not work. -5 DOES work.
We know that 0, numbers between 0 and 1, and numbers between 0 and -1 do not work with the statement.
I think if you had carried this line of thought a little further, you would have arrived at
what we can conclude with logic in just a few seconds:
if |x| < x^2, \(x>1\) or \(x<-1\)
Quote:
\(|x|≥0\)
This expression got you mixed up, I think.
For this problem, that equation works best to remind us that we can
divide the terms of the
given inequality by |x|.
Normally, we do not divide sides of inequalities by variables,
because we do not know whether the variables are positive or negative.
In this case, that equation tells us: the absolute value of \(x\) is either 0 or positive.
And in this case, \(x \neq 0.\) (See where we tested numbers.)
So if x cannot be 0, it MUST be positive.
In that case, we CAN divide by sides by \(|x|\)
When we did, we got
\(|x| < x^2\)
\(|x| < |x|*|x|\)
Divide both sides by |x|
\(\frac{|x|}{|x|}<\frac{|x|*|x|}{|x|}\)
\(1<|x|\), which is equivalent to
\(|x|>1\)
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