Which of the following could be the value of x, if |12 − 4x| + 2 = 6?

Question

A. −4
B. −2
C. 1
D. 3
E. 4

Kinshook · Answer

Asked: Which of the following could be the value of x, if |12 − 4x| + 2 = 6?

|12 − 4x| + 2 = 6
4|x-3| = 4
|x-3| = 1
x = 3 + -1
x = 2 or 4

IMO E

BrentGMATPrepNow · Answer

Take: |12 − 4x| + 2 = 6
Isolate the absolute value part by subtracting 2 from both sides of the inequality: |12 − 4x| = 4

NICE RULE:   If |x| = k, then x = k or x = -k

We get:  12 − 4x = 4  or  12 − 4x = -4

If  12 − 4x = 4 , then x = 2.
Check the answer choices....x = 2 is not an option. Keep going.

If  12 − 4x = -4 , then x = 4
Check the answer choices....bingo!

Answer: E

Cheers,
Brent

ScottTargetTestPrep · Answer

Solution:

Looking at the choices, we see that if x = 4, we have |12 - 16| + 2 = |-4| + 2 = 4 + 2 = 6.

(Note: We can solve this equation algebraically; however, sometimes it’s easier to just substitute in the numbers.)

Alternate Solution:

Let’s do the algebraic solution. First, we re-express the equation as |12 − 4x| = 4

Now, recall that to solve an absolute value equation, we always have two cases, as shown below.

Case 1.  Drop the absolute value signs and solve for x:

12 - 4x = 4

-4x = -8

x = 2

Case 2.  Drop the absolute value signs and negate the expression. Then solve for x.

-(12 - 4x) = 4

-12 + 4x = 4

4x = 16

x = 4.

Of the two solutions x = 2 and x = 4, only x = 4 appears in the answer choices.

Answer: E

BrushMyQuant · Answer

Given that |12 − 4x| + 2 = 6 and we need to find which value of x out of the option choices satisfies this.

Let's understand how to solve this problem using two methods

Method 1: Substitution

Let's take each answer choice and substitute in the question and check which one satisfies the question

A. −4  Put x = -4 in |12 − 4x| + 2 = 6. We get
| 12 - 4*(-4)| + 2 = | 12 + 16| + 2 = |28| + 2 = 28 + 2 = 30  
eq  6 =>   NOT POSSIBLE

B. −2  Put x = -2 in |12 − 4x| + 2 = 6. We get
| 12 - 4*(-2)| + 2 = | 12 + 8| + 2 = |20| + 2 = 20 + 2 = 22  
eq  6 =>   NOT POSSIBLE

C. 1  Put x = 1 in |12 − 4x| + 2 = 6. We get
| 12 - 4*1| + 2 = | 12 - 4 | + 2 = |8| + 2 = 8 + 2 = 10  
eq  6 =>   NOT POSSIBLE

D. 3  Put x = 3 in |12 − 4x| + 2 = 6. We get
| 12 - 4*3| + 2 = | 12 - 12| + 2 = |0| + 2 = 0 + 2 = 2  
eq  6 =>   NOT POSSIBLE

E. 4  Put x = 4 in |12 − 4x| + 2 = 6. We get
| 12 - 4*4| + 2 = | 12 - 16| + 2 = |-4| + 2 = 4 + 2 = 6 = 6 =>   POSSIBLE

Method 2: Algebra

|12 − 4x| + 2 = 6 
=> |12 − 4x| = 6 -2 = 4
Divide both the sides by 4 we get
 \frac{|12-4x|}{4} = \frac{4}{4} 
=> | 3 - x | = 1

Now, there are  two ways  of solving this

Method 2.1: Squaring

Square both the sides we get 
 (|3-x|)^2 = 1^2 
=>  (3-x) ^ 2  = 1
=>  3^2 + x^2 - 2*3*x  = 1
=> 9 +  x^2  - 6x - 1 = 0
=>  x^2  - 6x + 8 = 0
=>  x^2  - 2x - 4x + 8 =0
=> x*(x-2) -4*(x-2) = 0
=> (x-2) * (x-4) = 0
=> x = 2 or x = 4

Method 2.2: Opening Absolute Value

| 3 - x | = 1 
=> 3-x = 1 or 3-x =-1
=> x = 3-1 = 2 or x = 3+1 = 4

So,  Answer will be E 
Hope it helps!

Watch the following video to MASTER Absolute Values

https://www.youtube.com/watch?v=huFiBk8PaGY

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Which of the following could be the value of x, if |12 − 4x| + 2 = 6?

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