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The Reason Behind Absolute Value Questions on the GMAT

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Re: The Reason Behind Absolute Value Questions on the GMAT [#permalink]
Bunuel wrote:
[rss2posts title="Veritas Prep Blog" title_url="https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2014/06/the-reason-behind-absolute-value-questions-on-the-gmat/" sub_title="The Reason Behind Absolute Value Questions on the GMAT"]

|x| = x if x >= 0

|x| = -x if x < 0

So you can substitute x for |x| to make it a regular equation but only if x is non negative. If x is negative, then you put -x instead of |x| to convert it into a simple equation. And that is the reason you need to take positive and negative values of what is inside the absolute value sign.

I still don't understand WHY |x| = -x when x<0.
To my understanding |x| is the distance of x from zero and distances are not negative. So, then why?
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The Reason Behind Absolute Value Questions on the GMAT [#permalink]
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Karmesh wrote:
Bunuel wrote:
[rss2posts title="Veritas Prep Blog" title_url="https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2014/06/the-reason-behind-absolute-value-questions-on-the-gmat/" sub_title="The Reason Behind Absolute Value Questions on the GMAT"]

|x| = x if x >= 0

|x| = -x if x < 0

So you can substitute x for |x| to make it a regular equation but only if x is non negative. If x is negative, then you put -x instead of |x| to convert it into a simple equation. And that is the reason you need to take positive and negative values of what is inside the absolute value sign.

I still don't understand WHY |x| = -x when x<0.
To my understanding |x| is the distance of x from zero and distances are not negative. So, then why?

What is l-7l, it is the distance of -7, on the number line, from zero.
No let's follow the formula lxl = -(x) iff x<0,
Here x=-7, cool.
so answer is - (-7), i.e. 7.

So, the absolute value (read distance from zero on the number line) of -7 is 7

Hope this helps. Karmesh

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Re: The Reason Behind Absolute Value Questions on the GMAT [#permalink]
Karmesh wrote:
Bunuel wrote:
[rss2posts title="Veritas Prep Blog" title_url="https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2014/06/the-reason-behind-absolute-value-questions-on-the-gmat/" sub_title="The Reason Behind Absolute Value Questions on the GMAT"]

|x| = x if x >= 0

|x| = -x if x < 0

So you can substitute x for |x| to make it a regular equation but only if x is non negative. If x is negative, then you put -x instead of |x| to convert it into a simple equation. And that is the reason you need to take positive and negative values of what is inside the absolute value sign.

I still don't understand WHY |x| = -x when x<0.
To my understanding |x| is the distance of x from zero and distances are not negative. So, then why?

You are ignoring the second part of the statement: "when x < 0"
When x itself is NEGATIVE, -x is POSITIVE. Of course, distance is positive. Hence when x itself is positive, |x| is x but when x itself is negative, you need to make it positive by making it -x.
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Re: The Reason Behind Absolute Value Questions on the GMAT [#permalink]
KarishmaB how to identify while opening these cases as to when to make them equal also given that the quest 2 has an inequality
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Re: The Reason Behind Absolute Value Questions on the GMAT [#permalink]
Bunuel wrote:

Question 2:  If x and y are integers and y = |x+3| + |4-x|, does y equal 7?

Statement 1: x < 4

Statement 2: x > -3

Solution: Now what do you do when you have y = |x+3| + |4-x|? How do you convert this into a regular equation? You don’t know whether whatever is in the absolute value sign is positive or negative. How will you get rid of the sign then? You will work on all the cases (messy algebra coming up!).

Now, we see the same logic in this question:

y = |x+3| + |4-x|

|x+3| = (x+3) if (x+3) >= 0. In other words, if x >= -3

|x+3| = -(x+3) if (x+3) < 0. In other words, if x < -3

|4-x| = (4-x) if (4-x) >= 0. In other words, if x <= 4

|4-x| = -(4-x) if (4-x) < 0. In other words, if x > 4

So our absolute values behave differently when x < -3, between -3 and 4 and when x > 4. We say that -3 and 4 are our transition points.

Case 1:

When x < -3, |x+3| = -(x+3) and |4-x| = (4-x).

So the equation becomes y = -(x+3) + (4-x)

y = 1 – 2x

For different values of x, y will take different values. Recall that x must be less than -3. Say x = -4, then y = 9. If x = -5, y = 11.

Case 2:

When -3 <= x <= 4, |x+3| = (x+3) and |4-x| = (4-x).

So the equation becomes y = (x+3) + (4-x)

y = 7

In this range, y will always be 7.

Case 3:

When x > 4, |x+3| = (x+3) and |4-x| = -(4-x)

So the equation becomes y = (x+3) – (4-x)

y = 2x – 1

For different values of x, y will take different values. Recall that x must be more than 4. Say x = 5, then y is 9. If x = 6, then y is 11.

Note that y equals 7 only when x is between -3 and 4. Both statements together tell us that x is between -3 and 4. No statement alone gives us this information. Hence, using both statements, we know that y must be 7.

From all the cases solved why does Y have to be 7. This is the part I do not understand. Someone plox help.
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Re: The Reason Behind Absolute Value Questions on the GMAT [#permalink]
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Re: The Reason Behind Absolute Value Questions on the GMAT [#permalink]
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