How do you solve $ 4\left| {7 - y} \right| - 1 = 11? $

Hint: Take the given expression and apply the property of the absolute value and simplify the equation giving plus or minus sign while removing the mode, then simplify the equation for the resultant value for “y”.

Complete step by step solution:
Take the given expression: $ 4\left| {7 - y} \right| - 1 = 11 $
Move the number from the left hand side of the equation to the right hand side of the equation. When you move any term from one side of the equation to the other the sign of the term also changes. Negative terms become positive and vice-versa.
 $ 4\left| {7 - y} \right| = 11 + 1 $
Simplify the above equation finding the sum of the terms on the right hand side of the equation.
 $ 4\left| {7 - y} \right| = 12 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ \left| {7 - y} \right| = \dfrac{{12}}{4} $
Find factors for the term on the numerator of the right hand side of the equation.
 $ \left| {7 - y} \right| = \dfrac{{4 \times 3}}{4} $
Common factors from the numerator and the denominator cancels each other. Therefore, remove from the numerator and the denominator of the above expression.
 $ \left| {7 - y} \right| = 3 $
Remove the mode in the above expression which gives
 $ 7 - y = \pm 3 $
Therefore, there are two values
 $ 7 - y = 3 $
Move the term “y” on the opposite side and constant on one side.
 $ 7 - 3 = y $
Simplify the above equation
 $ 4 = y $
Therefore, $ y = 4 $ …. (A)
 $ 7 - y = - 3 $
Move the term “y” on the opposite side and constant on one side.
 $ 7 + 3 = y $
Simplify the above equation
 $ 10 = y $
Therefore, $ y = 10 $ …. (B)
Hence, the resultant answer is $ y = 4,10 $
So, the correct answer is “ $ y = 4,10 $ ”.

Note: Always remember when you remove mode sign, there will be plus or minus sign for the equivalent value of mode. Be careful in sign convention while moving any term from one side to another.