Question

complete values of x satisfying |x+5|+|6-x|=11 is

A. (-6, 5)
B. (5, 6)
C. (-5, 6)
D. (-5, -6)

Please answer immediately with method.​

Answer 1

Given : |x+5|+|6-x|=11  

To Find :   Values of x satisfying

Solution :

|x+5|+|6-x|=11  

| x |  =  x   if  x ≥ 0

      = - x  if x  <  0

Case 1  :  x  >  6    =>  6  - x   <  0    and x + 5 >  0

=>  x  + 5  -  (6 - x)  = 11

=>  2x - 1  = 11

=> 2x = 12

=> x  =  6

Hence no solution for  x > 6

Case 2  : x  < - 5     => x + 5 <  0      6 - x > 0

Hence

-(x + 5)  + 6 - x  =  11

=> - 2x  + 1 = 11

=> x = - 5  

Hence no solution   for x < - 5

Case 3 :     -5 ≤ x  ≤ 6

x + 5 ≥ 0   ,  6 - x ≥  0

=>  x + 5  + 6 - x  =  11

=> 11  = 11

Satisfied for all values of x

Hence x   ∈ [-5 , 6]

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Answer 2

SOLUTION

TO CHOOSE THE CORRECT OPTION

Complete values of x satisfying

$\sf{ |x + 5| + |6 - x| = 11 }$

A. [-6, 5]

B. [5, 6]

C. [-5, 6]

D. [-6, -5]

EVALUATION

Here the given equation is

$\sf{ |x + 5| + |6 - x| = 11 }$

Here

| x + 5 | = 0 when x = - 5

| 6 - x | = 0 when x = 6

Case : 1

When x < - 5

$\sf{ |x + 5| + |6 - x| }$

$\sf{ = - (x + 5)+ (6 - x) }$

$\sf{ = - x - 5+ 6 - x }$

$\sf{ = 1 - 2x }$

Case : 2

When - 5 ≤ x ≤ 6

$\sf{ |x + 5| + |6 - x| }$

$\sf{ = (x + 5) + (6 - x) }$

$\sf{ = x + 5+ 6 - x }$

$\sf{ = 11}$

Case : 3

When x > 6

$\sf{ |x + 5| + |6 - x| }$

$\sf{ = (x + 5) - (6 - x) }$

$\sf{ = x + 5- 6 + x }$

$\sf{ = 2x - 1 }$

Above three cases can rewritten as

$\sf {} |x + 5| + |6 - x| = \begin{cases} & \sf{ \: \: \: \: 1 - 2x \: \: \: \: \: \: \: \: \: when \: x < - 5} \\ \\& \sf{ \: \: \: \: \: \: \: \: \: 11 \: \: \: \: \: \: \: \: \: \: \: \:when \: -5 \leqslant x \leqslant 6} \\ \\ & \sf{ \: \: \: \: \: 2x - 1 \: \: \: \: \: \: \: \: when \: x > 6} \end{cases}$

$\sf{Hence \: \: |x + 5| + |6 - x| = 11 \: \: holds \: \: when \: x \in \: [-5,6] }$

FINAL ANSWER

Hence the correct option is C. [ - 5 , 6 ]

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