What is the value of │x + 7│?

Question: What is the value of │x + 7│?

(1) │x + 3│= 14

(2) (x + 2)^2 = 169

1. 12
2. 10
3. 11
4. 14

Solution and Explanation: 

Approach Solution 1:

The solution can be achieved with two different approaches. First assuming |x+3| = 14. This assumption will give two probable outcomes; x+3 = 14 and x+3 = -14.

This implies x = 11, -17.

The mentioned value is not sufficient for the values mentioned above. Thus, this can’t be the solution.

The second approach is (x+2)^2 = 169. Thus, this implies x+2 = +-13. Therefore, the probable values of x are -15, and 11.

Now, comparing both the assumptions, there is a single value of x that satisfies both the cases i.e x=11. Therefore, the answer will be x=11.

Approach Solution 2:

Another approach that can be adapted by candidate are following these three steps:

• Apply the rule that says: If |x| = k, then x = k and/or x = -k
• Solve the resulting equations
• Plug in the solutions to check for extraneous roots

So for the above mentioned question, x+3 = 14 Or x+3 = -14. Upon evaluation, we get two values of x, 11 and -17.

The next approach is to input the value of x within the given equations in the question. Thus, we get;

If x = 11, then |x + 7| = |11 + 7| = 18
If x = -17, then |x + 7| = |-17 + 7| = 10

Since, it doesn’t satisfy the value in the question, thus, this value is not sufficient.

Now, using the second equation mentioned in the question, (x+2)² = 169 implies (x+2) = 13 OR (x+2) = -13. Upon solving the equation, we get x = 11 OR x = -15. Now putting the value of x in the given equation |x + 7|, we get;

if x = 11, then |x + 7| = |11 + 7| = 18
if x = -15, then |x + 7| = |-15 + 7| = 8

Combining both the equations, |x + 7| = 18 OR 10 and |x + 7| = 18 OR 8, derives the conclusion that |x + 7| must equal 18. Thus, upon solving |x + 7| = 18, the value x= 11. Therefore, the value of x=11.

Approach Solution 3:

Statement 1

In order to solve this, we need to consider two different cases.

Case 1: x + 3 > 0

Thus, if x + 3 > 0 then the value of |x+3| will be positive. This implies x+3 = 14, thus, x= 14-13

Further, x=11.

Now putting the value of x=11 in x+3=14 we get 11+3=14, that justifies the case 14 > 0.

Case 2: Consider x+3 < 0

Thus, this implies that the value of |x+3| will be negative.

Therefore, the equation formed will be -(x+3) = 14,

=> x= -14-3
=> x= -17.

Now putting the value of x=-17 in x+3=14 we get -17+3=14, that justifies the case -14 < 0.

Two different values => x alone is not sufficient

Statement 2

The second statement in the question states that (x + 2)^2 = 169

Since, \(\sqrt{(x^2)}=|x|\), thus it can be stated that |x+2|=13.

Case 1: x+2>0

It can be concluded from statement 1 that value of |x+2| will be positive and hence it will be x+2=13 => x=11.

Putting the value of x in |x+2|=13, |11+2|=13, thus, it is justified that x=11.

Case 2: x+2<0

As mentioned in statement 1, it can be stated that the value of |x+2| will be negative and hence it will be -(x+2)

Implies; -x-2=13
Therefore, x=-15

Putting the value of x in -|x+2|=13, -15+2=-13, thus, it is justified that x=-15 is justified.

Combining both the statements, we get a common value x=11, thus, the answer will be 11.

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