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When is x4 = 4x? [#permalink]
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03 May 2012, 11:38
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When is x4 = 4x? A. x=4 B. x=0 C. x>4 D. x<=4 E. x< 0 I could answer this question by plugging in some numbers. But how do i prove this using algebra?
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Re: When is x4 = 4x? [#permalink]
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08 May 2012, 10:57
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nkimidi7y wrote: 4x is always >=0. So x is always <=4.
Is this what you have meant? x4 = 4x 1) x4 is ALWAYS none negative. so , 4x must be also more or equal to zero 4x>=0 x<=4 2) if x<4 , then 4x=4x if x>=4 x4=4x x=4 so x <= 4
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Re: When is x4 = 4x? [#permalink]
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4x is always >=0. So x is always <=4.
Is this what you have meant?



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Re: When is x4 = 4x? [#permalink]
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Absolute value of any number or expression must be positive. If (x4) is positive then x4 is also positive What if x4 is negative? Since the absolute value must be positive, x4 would be equal to (x4)=4x. Right? We know that x4 would have to be negative for the equation in question to be true. This would imply that x would have to be a small positive number smaller than 4 or a negative number. You can take examples to test that. x=14 (x4)=ve x=1, x4=3 ve x=4, implies x4=0 and 4x=0. Thus, the equation is satisfied. Hence, d is the answer. Coming to your question, if a question deals with equality it also indirectly deals with inequality. If you say the equation is satisfied when x=0,x=4,x=5 and so on, it also implies that the equation is true for all values of x less than or equal to 4. An equation exists only at certain points. We have to find those points and if those points range over a large space, the easiest way would be express it as inequality. Note: An equality question can have answers which might be expressed as inequalities. There is nothing wrong with it. Hope it helps! Let me know if I can help you any further. dhlee922 wrote: how come all of a sudden the answer has inequalities when the question only had equal signs? that's the part i dont understand



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Re: When is x4 = 4x? [#permalink]
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mario1987 wrote: Bunuel wrote: nkimidi7y wrote: When is x4 = 4x?
A. x=4 B. x=0 C. x>4 D. x<=4 E. x< 0
I could answer this question by plugging in some numbers. But how do i prove this using algebra? Absolute value properties:When \(x\leq{0}\) then \(x=x\), or more generally when \(some \ expression\leq{0}\) then \(some \ expression\leq{(some \ expression)}\). For example: \(5=5=(5)\); When \(x\geq{0}\) then \(x=x\), or more generally when \(some \ expression\geq{0}\) then \(some \ expression\leq{some \ expression}\). For example: \(5=5\); So, \(x4=4x=(x4)\) to be true should be that \(x4\leq{0}\) > \(x\leq{4}\). Answer: D. Hope it's clear. Hi Bunuel I am trying to understand theese two properties, but how is it possible to have X=X, in order that absolute value has to be always positive? Could you please provide me an explaination in more details? Thanks in advance When \(x\leq{0}\), for example when \(x=5\), then \(5=5=(5)\) so \(x=x\) (negative =(negative)=positive). For more check Absolute Value chapter of Math book: mathabsolutevaluemodulus86462.htmlHope it helps.
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Re: When is x4 = 4x? [#permalink]
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When is x4 = 4x?
Critical Values method:
The Critical Value here is x=4 (we make the absolute value term equal to zero), so we have this conditions to check:
1) x<4: (x4)=4x > x4=x4 > true for all values of x, but only when x<4 the initial condition is satisfied > true always that x<4
2) x=4: 0=0 > this is true always that x=4
3) x>4: x4=4x > 2x=8 > x=4 > initial condition of x>4 is not met
Therefore, there is a solution only when: x<=4
Solution D



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Re: When is x4 = 4x? [#permalink]
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29 Nov 2014, 00:23
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Question asks when distance between x and 4 expressed as x4 is equal to 4x
until x is left to 4 or equal to 4 we get the equation right so x4=4x
Number line is
x4>
if x goes righter we get
4x>
x4 will continue to be positive, but 4x will be negative
Algebraically:
x4=4x 2x=8, x=4
(x4)=4x x+4=4x 0=0, so infinitely many solutions when x<4
x<=4
D



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Re: When is x4 = 4x? [#permalink]
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28 Oct 2012, 02:56
Bunuel wrote: nkimidi7y wrote: When is x4 = 4x?
A. x=4 B. x=0 C. x>4 D. x<=4 E. x< 0
I could answer this question by plugging in some numbers. But how do i prove this using algebra? Absolute value properties:When \(x\leq{0}\) then \(x=x\), or more generally when \(some \ expression\leq{0}\) then \(some \ expression\leq{(some \ expression)}\). For example: \(5=5=(5)\); When \(x\geq{0}\) then \(x=x\), or more generally when \(some \ expression\geq{0}\) then \(some \ expression\leq{some \ expression}\). For example: \(5=5\); So, \(x4=4x=(x4)\) to be true should be that \(x4\leq{0}\) > \(x\leq{4}\). Answer: D. Hope it's clear. Hi Bunuel I am trying to understand theese two properties, but how is it possible to have X=X, in order that absolute value has to be always positive? Could you please provide me an explaination in more details? Thanks in advance



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Re: When is x4 = 4x? [#permalink]
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05 Dec 2012, 04:51
x4 = 4x Absolute values are always positive or greater than 0. So, 4x >= 0 ==> x <= 4 Answer: D There is no need to test values.
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Re: When is x4 = 4x? [#permalink]
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06 Jan 2013, 14:26
Hi brunel,
You said some expression when it is > = 0 then l some espression l <= some expression.
In this case how do i know X4 is >= 0
Thanks!



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Re: When is x4 = 4x? [#permalink]
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10 Feb 2013, 06:14
Why not E?
If i am not mistaken when x<0, then x1 = x1?



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Re: When is x4 = 4x? [#permalink]
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Re: When is x4 = 4x? [#permalink]
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13 Mar 2013, 14:45
how come all of a sudden the answer has inequalities when the question only had equal signs? that's the part i dont understand



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Re: When is x4 = 4x? [#permalink]
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14 Mar 2013, 17:34
thanks Kris. that does help. i guess my follow up question would be, is there a way to solve it algebraically rather than plugging in numbers? Kris01 wrote: Absolute value of any number or expression must be positive. If (x4) is positive then x4 is also positive What if x4 is negative? Since the absolute value must be positive, x4 would be equal to (x4)=4x. Right? We know that x4 would have to be negative for the equation in question to be true. This would imply that x would have to be a small positive number smaller than 4 or a negative number. You can take examples to test that. x=14 (x4)=ve x=1, x4=3 ve x=4, implies x4=0 and 4x=0. Thus, the equation is satisfied. Hence, d is the answer. Coming to your question, if a question deals with equality it also indirectly deals with inequality. If you say the equation is satisfied when x=0,x=4,x=5 and so on, it also implies that the equation is true for all values of x less than or equal to 4. An equation exists only at certain points. We have to find those points and if those points range over a large space, the easiest way would be express it as inequality. Note: An equality question can have answers which might be expressed as inequalities. There is nothing wrong with it. Hope it helps! Let me know if I can help you any further. dhlee922 wrote: how come all of a sudden the answer has inequalities when the question only had equal signs? that's the part i dont understand



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Re: When is x4 = 4x? [#permalink]
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15 Mar 2013, 11:26
As Johnwesley said, for x4=4x, s4 should be negative or equal to 0.
i.e. x4<=0 Hence, x<=4
[quote="dhlee922"]thanks Kris. that does help. i guess my follow up question would be, is there a way to solve it algebraically rather than plugging in numbers?



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Re: When is x4 = 4x? [#permalink]
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09 Aug 2013, 22:24
I solved it in this way.
When is x4 = 4x?
Choice A: X=4, it is true but X cannot be always 4 Choice B: X=0, it is also true, but X cannot be always 0 Choice C: X>4, it is false, for e.g. X=6, then one side of equation is 2 and the other side is 2 Choice D: X<=4, this choice encapsulate Choice A, Choice B and for all other conditions and is true for above said equation. Hence the answer choice is D.
It took only 1min to solve this problem with above method.
A. x=4 B. x=0 C. x>4 D. x<=4 E. x< 0



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Re: When is x4 = 4x? [#permalink]
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10 Aug 2013, 14:15
nkimidi7y wrote: When is x4 = 4x?
A. x=4 B. x=0 C. x>4 D. x<=4 E. x< 0
I could answer this question by plugging in some numbers. But how do i prove this using algebra? ..... picking numbers is best for this sort of problem. you did well already..... it saves time.....
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Re: When is x4 = 4x? [#permalink]
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19 Dec 2013, 05:14
Bunuel wrote: Victorlp89 wrote: Why not E?
If i am not mistaken when x<0, then x1 = x1? First of all we have x4 = 4x not x1 = x1. Next, x1 = x1 for \(x\geq{1}\). If \(x\leq{1}\), then \(x1=(x1)=1x\) Hi Bunuel, Sinetunes you can manipulate say x4=(x+4)=4x is what you'll end up with. So then how do you solve when you have 4x = 4x? Is this property correct or what are its limitations? Thanks J




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