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When is |x-4| = 4-x? [#permalink]
 03 May 2012, 11:38
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When is |x-4| = 4-x? 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 |x-4| = 4-x? [#permalink]
03 May 2012, 12:04
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nkimidi7y wrote: When is |x-4| = 4-x?
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, |x-4|=4-x=-(x-4) to be true should be that x-4\leq{0} --> x\leq{4}. Answer: D. Hope it's clear.
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Re: When is |x-4| = 4-x? [#permalink]
03 May 2012, 15:00
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4-x is always >=0.
So x is always <=4.
Is this what you have meant?
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Re: When is |x-4| = 4-x? [#permalink]
08 May 2012, 10:57
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nkimidi7y wrote: 4-x is always >=0.
So x is always <=4.
Is this what you have meant? |x-4| = 4-x 1) |x-4| is ALWAYS none negative. so , 4-x must be also more or equal to zero 4-x>=0 x<=4 2) if x<4 , then 4-x=4-x if x>=4 x-4=4-x x=4 so x <= 4
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Re: When is |x-4| = 4-x? [#permalink]
28 Oct 2012, 02:56
Bunuel wrote: nkimidi7y wrote: When is |x-4| = 4-x?
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, |x-4|=4-x=-(x-4) to be true should be that x-4\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 |x-4| = 4-x? [#permalink]
29 Oct 2012, 02:56
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mario1987 wrote: Bunuel wrote: nkimidi7y wrote: When is |x-4| = 4-x?
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, |x-4|=4-x=-(x-4) to be true should be that x-4\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: math-absolute-value-modulus-86462.htmlHope it helps.
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Re: When is |x-4| = 4-x? [#permalink]
05 Dec 2012, 04:51
|x-4| = 4-x
Absolute values are always positive or greater than 0.
So, 4-x >= 0 ==> x <= 4
Answer: D
There is no need to test values.
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Re: When is |x-4| = 4-x? [#permalink]
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 X-4 is >= 0
Thanks!
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Re: When is |x-4| = 4-x? [#permalink]
07 Jan 2013, 03:37
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Re: When is |x-4| = 4-x? [#permalink]
10 Feb 2013, 06:14
Why not E?
If i am not mistaken when x<0, then |x-1| = x-1?
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Re: When is |x-4| = 4-x? [#permalink]
10 Feb 2013, 06:21
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Re: When is |x-4| = 4-x? [#permalink]
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 |x-4| = 4-x? [#permalink]
14 Mar 2013, 13:19
Absolute value of any number or expression must be positive. If (x-4) is positive then |x-4| is also positive What if x-4 is negative? Since the absolute value must be positive, |x-4| would be equal to -(x-4)=4-x. Right? We know that x-4 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 (x-4)=-ve x=1, x-4=-3 -ve x=4, implies x-4=0 and 4-x=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 |x-4| = 4-x? [#permalink]
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 (x-4) is positive then |x-4| is also positive What if x-4 is negative? Since the absolute value must be positive, |x-4| would be equal to -(x-4)=4-x. Right? We know that x-4 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 (x-4)=-ve x=1, x-4=-3 -ve x=4, implies x-4=0 and 4-x=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 |x-4| = 4-x? [#permalink]
14 Mar 2013, 18:19
When is |x-4| = 4-x? 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: -(x-4)=4-x ---> x-4=x-4 ---> 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: x-4=4-x ---> 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 |x-4| = 4-x? [#permalink]
15 Mar 2013, 11:26
As Johnwesley said, for |x-4|=4-x, s-4 should be negative or equal to 0. i.e. x-4<=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 |x-4| = 4-x?
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15 Mar 2013, 11:26
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