How many solutions does the equation |x−7|+|2x−4|+ x = 6 have ? A) non

Question

How many solutions does the equation  |x−7|+|2x−4|+ x = 6  have ?

A) none

B) one

C) two

D) three

E) four

ThatDudeKnows · Accepted Answer

nislam

Great absolute value question! If you ABSOLUTEly (see what I did there?) nail your understanding of this question to the point where you can do and explain it in your sleep, you should be set for the topic!

We have "break points" at 7 and 2, at which one of the absolute values would switch from positive to negative.

If x>=7, x-7 is non-negative and 2x-4 is positive. Since both of the quantities inside the absolute value bars are non-negative, we can ignore the absolute value bars.
 (x−7)+(2x−4)+ x = 6 
 4x−11 = 6 
 4x = 17 
 x = 4.25 
But we started with "if x>=7," so x can't be 4.25. Therefore, there is no solution with x>=7.

If 7>x>=2, x-7 is negative and 2x-4 is non-negative. Since x-7 is negative, we need to take the negative of that quantity. Since 2x-4 is positive, we can ignore the absolute value bars. 
 -(x−7)+(2x−4)+ x = 6 
 -x+7+2x−4+x = 6 
 2x+3 = 6 
 2x = 3 
 x = 1.5 
But we started with "if 7>x>=2," so x can't be 1.5. Therefore, there is no solution with 7>x>=2.

If 2>x, x-7 is negative and 2x-4 is negative. Since both quantities are negative, we need to take the negative of each.
 -(x−7)-(2x−4)+ x = 6 
 -x+7-2x+4+x = 6 
 -2x+11 = 6 
 -2x = -5 
 x = 2.5 
But we started with "if 2>x," so x can't be 2.5. Therefore, there is no solution with 2>x.

We've shown that there is no solution where x is greater than 7, no solution where x is between 2 and 7, and no solution when x is less than 2. There is no solution.

Answer choice A.

lovikansal · Answer

Had it been the case that the question required integral values then none would have been correct.
Can you please explain why there are no solutions?

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nislam · Answer

could anyone please give the detail solution of the problem?

av1901 · Answer

That's the catch with modulus questions,you always got to check back the solutions with the original equation
If you do, you will see that none of the solutions/values we got satisfy the equation

That is why no possible solution

BrushMyQuant · Answer

|x−7|+|2x−4|+ x = 6

We have two absolute values so let's find out the points on the number line where sign of either absolute value term will change
=>x - 7 = 0 or 2x - 4 = 0=> x = 7 or x = 2
   Let's divide the number line into three parts now:  
 
   -   Case 1:  x ≤ 2
 
=> Both x - 7 and 2x - 4 will be non-positive
=> |x - 7| = -(x - 7) = 7 - x and 
=> |2x-4| = -(2x - 4) = 4 - 2x
=> 7 - x + 4 - 2x + x = 6
=> 2x = 11 - 6 = 5
=> x = 5/2 = 2.5

But condition was x ≤ 2  
=>  x = 2.5 is  NOT a SOLUTION   
  -   Case 2:  2 ≤ x ≤ 7
 
=> x - 7 will be non-positive and 2x - 4 will be non-negative
=> |x - 7| = -(x - 7) = 7 - x and 
=> |2x-4| = 2x - 4
=> 7 - x + 2x - 4 + x = 6
=> 2x = 3
=> x = 3/2 = 1.5

But condition was 2 ≤ x ≤ 7  
=>  x = 1.5 is  NOT a SOLUTION    
  -   Case 3:  x  ≥ 7
  
 => Both x - 7 and 2x - 4 will be non-negative
=> |x - 7| = x - 7 and 
=> |2x-4| = 2x - 4
=> x - 7 + 2x - 4 + x = 6
=> 4x = 11 + 6 + 4 = 21
=> x = 21/4 = 5.25

But condition was x  ≥ 7   
=>  x = 5.25 is  NOT a SOLUTION

=>  There is  no solution  to above equation

So,  Answer will be A 
Hope it helps!

Watch the following video to MASTER Absolute Value Problems

https://www.youtube.com/watch?v=rBWfl4AmQDI

Rohanx9 · Answer

If you don’t want to do piecewise equations, here’s a slicker approach using plugging in values to figure out inflection points and slope logic.

The kinks (inflection points) are at x=7 and x=2. Three intervals:

x < 2
x = 0: |0−7| + |0−4| + 0 = 7 + 4 = 11
x = 2: |2−7| + |4−4| + 2 = 5 + 0 + 2 = 7
So the function drops from 11 at x=0 to 7 at x=2. This means the function has a negative slope on [0,2].

2 ≤ x < 7
x = 5: |5−7| + |10−4| + 5 = 2 + 6 + 5 = 13
It increases again by x=5. So on [2,7] the function has positive slope.

x ≥ 7
x = 7: |7−7| + |14−4| + 7 = 0 + 10 + 7 = 17
x = 10: |10−7| + |20−4| +10 = 3 + 16 +10 = 29
So beyond x=7 it continues to rise, meaning the function has positive slope on [7,∞)

The lowest value occurs at x=2, where the function equals 7. All slopes are positive. Since 7 > 6, the function never drops to 6, and keeps increasing after.

So there are no solutions. x = 10: |10−7| + |20−4| +10 = 3 + 16 +10 = 29

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Let's divide the number line into three parts now:
-Case 1: x ≤ 2 => Both x - 7 and 2x - 4 will be non-positive => \|x - 7\| = -(x - 7) = 7 - x and => \|2x-4\| = -(2x - 4) = 4 - 2x => 7 - x + 4 - 2x + x = 6 => 2x = 11 - 6 = 5 => x = 5/2 = 2.5 But condition was x ≤ 2 => x = 2.5 is NOT a SOLUTION	-Case 2: 2 ≤ x ≤ 7 => x - 7 will be non-positive and 2x - 4 will be non-negative => \|x - 7\| = -(x - 7) = 7 - x and => \|2x-4\| = 2x - 4 => 7 - x + 2x - 4 + x = 6 => 2x = 3 => x = 3/2 = 1.5 But condition was 2 ≤ x ≤ 7 => x = 1.5 is NOT a SOLUTION	-Case 3: x ≥ 7 => Both x - 7 and 2x - 4 will be non-negative => \|x - 7\| = x - 7 and => \|2x-4\| = 2x - 4 => x - 7 + 2x - 4 + x = 6 => 4x = 11 + 6 + 4 = 21 => x = 21/4 = 5.25 But condition was x ≥ 7 => x = 5.25 is NOT a SOLUTION

How many solutions does the equation |x−7|+|2x−4|+ x = 6 have ? A) non

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How many solutions does the equation |x−7|+|2x−4|+ x = 6 have ? A) non

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