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What is the minimum value of |x +11| - |x - 7|?

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Joined: 21 Feb 2017
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What is the minimum value of |x +11| - |x - 7|?  [#permalink]

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07 May 2020, 08:09
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What is the minimum value of |x +11| - |x - 7|?

A. -18
B. -4
C. 0
D. 4
E. 18

@Experts how is the answer -18? if i take x = -11 |-11+11| - |-11-7| => - |-18| => value inside the mod is <0 therefore -18 will come out when i remove the mod -(-18) = +18
could someone correct what im doing wrong?
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Re: What is the minimum value of |x +11| - |x - 7|?  [#permalink]

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07 May 2020, 08:28
1
1
Kritisood wrote:
What is the minimum value of |x +11| - |x - 7|?

A. -18
B. -4
C. 0
D. 4
E. 18

@Experts how is the answer -18? if i take x = -11 |-11+11| - |-11-7| => - |-18| => value inside the mod is <0 therefore -18 will come out when i remove the mod -(-18) = +18
could someone correct what im doing wrong?

Kritisood

|x +11| - |x - 7| = (Distance of x from -11) - (Distance of x from +7)

So if you take any values which is -11 or less than -11 i.e. to the left of -11 on number line then the distance of value x from 7 will be maximum which will minimize the value of function

For any values of x ≤-11

then the distance |x +11| - |x - 7| = -18

e.g. @x = -12, |x +11| - |x - 7| = |-12 +11| - |-12 - 7| = 1-19 = -18
e.g. @x = -13, |x +11| - |x - 7| = |-13 +11| - |-13 - 7| = 2-20 = -18

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Re: What is the minimum value of |x +11| - |x - 7|?  [#permalink]

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07 May 2020, 08:30
1
1
Kritisood wrote:
What is the minimum value of |x +11| - |x - 7|?

A. -18
B. -4
C. 0
D. 4
E. 18

@Experts how is the answer -18? if i take x = -11 |-11+11| - |-11-7| => - |-18| => value inside the mod is <0 therefore -18 will come out when i remove the mod -(-18) = +18
could someone correct what im doing wrong?

Critical points of |x +11| - |x - 7| are -11 and 7.

When $$x < -11$$, then $$x + 11 < 0$$ and $$x - 7 < 0$$, which means that for this range $$|x +11| = -(x + 11)$$ and $$|x - 7| = -(x - 7) = 7 - x$$. So, in this range $$|x +11| - |x - 7| = -(x + 11) - (7 - x) = -18$$.

When $$-11 \leq x \leq 7,$$ then $$x + 11 \geq 0$$ and $$x - 7 \leq 0$$, which means that for this range $$|x +11| = x + 11$$ and $$|x - 7| = -(x - 7) = 7 - x$$. So, in this range $$|x +11| - |x - 7| = x + 11 - (7 - x) = 2x + 4$$. The least value for $$2x + 7$$ for given range is when $$x = -11$$, so the lowest value is -18.

When $$x > 7$$, then $$x + 11 > 0$$ and $$x - 7 > 0$$, which means that for this range $$|x +11| = x + 11$$ and $$|x - 7| = x - 7$$. So, in this range $$|x +11| - |x - 7| = x + 11) - (x - 7) = 18$$.

The graph of |x +11| - |x - 7| is given below:

Quote:
@Experts how is the answer -18? if i take x = -11 |-11+11| - |-11-7| => - |-18| => value inside the mod is <0 therefore -18 will come out when i remove the mod -(-18) = +18
could someone correct what im doing wrong?

If $$x = -11$$, $$|x +11| - |x - 7| = |-11 +11| - |-11 - 7| = |0| - |-18| = 0 - 18 = -18$$. (|-18| is 18)

The way you are doing: $$- |-18| = -(-(-18)) = -18$$.

Attachment:

Untitled.png [ 5.12 KiB | Viewed 1199 times ]

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Re: What is the minimum value of |x +11| - |x - 7|?  [#permalink]

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16 May 2020, 05:49
1
1
Kritisood wrote:
What is the minimum value of |x +11| - |x - 7|?

A. -18
B. -4
C. 0
D. 4
E. 18

@Experts how is the answer -18? if i take x = -11 |-11+11| - |-11-7| => - |-18| => value inside the mod is <0 therefore -18 will come out when i remove the mod -(-18) = +18
could someone correct what im doing wrong?

Case - I

$$(x +11) - (x - 7) = 18$$

Case - II

$$(- x - 11) - ( - x + 7 ) = -18$$, Hence, IMHO (A)
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Re: What is the minimum value of |x +11| - |x - 7|?  [#permalink]

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16 May 2020, 06:08
1
Mck2023 wrote:
Bunuel wrote:
Kritisood wrote:
What is the minimum value of |x +11| - |x - 7|?

A. -18
B. -4
C. 0
D. 4
E. 18

@Experts how is the answer -18? if i take x = -11 |-11+11| - |-11-7| => - |-18| => value inside the mod is <0 therefore -18 will come out when i remove the mod -(-18) = +18
could someone correct what im doing wrong?

Critical points of |x +11| - |x - 7| are -11 and 7.

When $$x < -11$$, then $$x + 11 < 0$$ and $$x - 7 < 0$$, which means that for this range $$|x +11| = -(x + 11)$$ and $$|x - 7| = -(x - 7) = 7 - x$$. So, in this range $$|x +11| - |x - 7| = -(x + 11) - (7 - x) = -18$$.

When $$-11 \leq x \leq 7,$$ then $$x + 11 \geq 0$$ and $$x - 7 \leq 0$$, which means that for this range $$|x +11| = x + 11$$ and $$|x - 7| = -(x - 7) = 7 - x$$. So, in this range $$|x +11| - |x - 7| = x + 11 - (7 - x) = 2x + 4$$. The least value for $$2x + 7$$ for given range is when $$x = -11$$, so the lowest value is -18.

When $$x > 7$$, then $$x + 11 > 0$$ and $$x - 7 > 0$$, which means that for this range $$|x +11| = x + 11$$ and $$|x - 7| = x - 7$$. So, in this range $$|x +11| - |x - 7| = x + 11) - (x - 7) = 18$$.

The graph of |x +11| - |x - 7| is given below:

Quote:
@Experts how is the answer -18? if i take x = -11 |-11+11| - |-11-7| => - |-18| => value inside the mod is <0 therefore -18 will come out when i remove the mod -(-18) = +18
could someone correct what im doing wrong?

If $$x = -11$$, $$|x +11| - |x - 7| = |-11 +11| - |-11 - 7| = |0| - |-18| = 0 - 18 = -18$$. (|-18| is 18)

The way you are doing: $$- |-18| = -(-(-18)) = -18$$.

Attachment:
Untitled.png

Hi Bunuel
I always find your quant explanation smooth,different from conventional ones and easy to understand. You are just brilliant.

However, I couldnt understand this explanatuon clearly. Actually i have some bad issues with Absolute Values Questions with equations as well inequalities. Can you suggest some best way to master them.
Thank you.

Posted from my mobile device

9. Inequalities

10. Absolute Value

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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What is the minimum value of |x +11| - |x - 7|?  [#permalink]

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07 May 2020, 08:35
Kritisood wrote:
What is the minimum value of |x +11| - |x - 7|?

A. -18
B. -4
C. 0
D. 4
E. 18

@Experts how is the answer -18? if i take x = -11 |-11+11| - |-11-7| => - |-18| => value inside the mod is <0 therefore -18 will come out when i remove the mod -(-18) = +18
could someone correct what im doing wrong?

Asked: What is the minimum value of |x +11| - |x - 7|?

|x +11| - |x - 7|

Case 1: x<-11
|x +11| - |x - 7| = -x-11 - (-x+7) = -11-7 = - 18
Case 2: -11<=x<=7
|x +11| - |x - 7| = x+11 - (-x+7) = 2x+4
For x=-11; 2x+4 = -22+4 = -18
Case 3: x>7
|x +11| - |x - 7| = x+11 - (x-7) = 11+7 = 18

Minimum value of |x +11| - |x - 7| = -18

IMO A
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Re: What is the minimum value of |x +11| - |x - 7|?  [#permalink]

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16 May 2020, 04:07
Bunuel wrote:
Kritisood wrote:
What is the minimum value of |x +11| - |x - 7|?

A. -18
B. -4
C. 0
D. 4
E. 18

@Experts how is the answer -18? if i take x = -11 |-11+11| - |-11-7| => - |-18| => value inside the mod is <0 therefore -18 will come out when i remove the mod -(-18) = +18
could someone correct what im doing wrong?

Critical points of |x +11| - |x - 7| are -11 and 7.

When $$x < -11$$, then $$x + 11 < 0$$ and $$x - 7 < 0$$, which means that for this range $$|x +11| = -(x + 11)$$ and $$|x - 7| = -(x - 7) = 7 - x$$. So, in this range $$|x +11| - |x - 7| = -(x + 11) - (7 - x) = -18$$.

When $$-11 \leq x \leq 7,$$ then $$x + 11 \geq 0$$ and $$x - 7 \leq 0$$, which means that for this range $$|x +11| = x + 11$$ and $$|x - 7| = -(x - 7) = 7 - x$$. So, in this range $$|x +11| - |x - 7| = x + 11 - (7 - x) = 2x + 4$$. The least value for $$2x + 7$$ for given range is when $$x = -11$$, so the lowest value is -18.

When $$x > 7$$, then $$x + 11 > 0$$ and $$x - 7 > 0$$, which means that for this range $$|x +11| = x + 11$$ and $$|x - 7| = x - 7$$. So, in this range $$|x +11| - |x - 7| = x + 11) - (x - 7) = 18$$.

The graph of |x +11| - |x - 7| is given below:

Quote:
@Experts how is the answer -18? if i take x = -11 |-11+11| - |-11-7| => - |-18| => value inside the mod is <0 therefore -18 will come out when i remove the mod -(-18) = +18
could someone correct what im doing wrong?

If $$x = -11$$, $$|x +11| - |x - 7| = |-11 +11| - |-11 - 7| = |0| - |-18| = 0 - 18 = -18$$. (|-18| is 18)

The way you are doing: $$- |-18| = -(-(-18)) = -18$$.

Attachment:
Untitled.png

Hi Bunuel
I always find your quant explanation smooth,different from conventional ones and easy to understand. You are just brilliant.

However, I couldnt understand this explanatuon clearly. Actually i have some bad issues with Absolute Values Questions with equations as well inequalities. Can you suggest some best way to master them.
Thank you.

Posted from my mobile device
Re: What is the minimum value of |x +11| - |x - 7|?   [#permalink] 16 May 2020, 04:07