Question

How do you solve $ - 4\left| {x + 5} \right| = - 16? $

Answer 1

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

Complete step by step solution:
Take the given expression: $ - 4\left| {x + 5} \right| = - 16 $
Negative sign from both the sides of the equation cancels each other and remove minus sign from both the sides of the equation.
 $ 4\left| {x + 5} \right| = 16 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ \left| {x + 5} \right| = \dfrac{{16}}{4} $
Find factors for the term on the numerator of the right hand side of the equation.
 $ \left| {x + 5} \right| = \dfrac{{4 \times 4}}{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| {x + 5} \right| = 4 $
Remove the mode in the above expression which gives
 $ x + 5 = \pm 4 $
Therefore, there are two values
I. $ x + 5 = 4 $
Make the term “x” the subject and move constant on the opposite side. When you move any term from one side of the equation to the other the sign of the term also changes. Positive terms become negative and vice-versa.
 $ x = 4 - 5 $
Simplify the above equation, when you simplify between one negative and one positive term do subtraction give sign of bigger number to the resultant value.
 $ x = - 1 $
Therefore, $ x = - 1 $ …. (A)
II. $ x + 5 = - 4 $
Make the term “x” the subject and move constant on the opposite side.
 $ x = - 4 - 5 $
Simplify the above equation, when you simplify between two negative terms you have to do addition and sign of negative to the resultant value
 $ x = - 9 $
Therefore, $ x = - 9 $ …. (B)
Hence, the resultant answer is $ x = - 1, - 9 $
So, the correct answer is “ $ x = - 1, - 9 $ ”.

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, also while simplifying between the like terms.