Question

How do you solve $ \left| {7 + 2x} \right| = 9 $ ?

Answer 1

Hint: First of all we will frame the two equations to get the absolute values from the equation given in the modulus and then will find the value for the unknown term “x”.

Complete step-by-step answer:
Take the given expression: $ \left| {7 + 2x} \right| = 9 $
Now remove mode and get the two equations-
 $ - (7 + 2x) = 9 $
   $ (7 + 2x) = 9 $
Now solve equations (A) and (B) one by one-
 $ - (7 + 2x) = 9 $
Open the brackets, remember when there is a negative sign outside the bracket then the sign of the term inside the bracket changes.
 $ - 7 - 2x = 9 $
Make like terms together. Move constants on one side and the term with variables on the other side. When you move any term from one side to another then the sign of the term also changes. Positive terms become negative and the negative term becomes positive.
 $ - 2x = 9 + 7 $
Simplify the above equation
 $ - 2x = 16 $
The above equation can be re-written as –
 $ 2x = - 16 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $
  x = - \dfrac{{16}}{2} \\
  x = ( - 8)\;{\text{ }} \ldots .{\text{ }}\left( C \right) \;
  $
 $ (7 + 2x) = 9 $
Make like terms together. Move constants on one side and the term with variables on the other side. . When you move any term from one side to another then the sign of the term also changes. Positive terms become negative and negative terms become positive.
 $ 2x = 9 - 7 $
Simplify the above equation
 $ 2x = 2 $
Make “X” the subject
 $ \Rightarrow x = \dfrac{2}{2} $
Common factors from the numerator and the denominator cancel each other.
 $ \Rightarrow x = 1 $ … (D)
Hence, the required solution is $ x = - 8 $ or $ x = 1 $
So, the correct answer is “ $ x = - 8 $ or $ x = 1 $ ”.

Note: Be careful about the sign convention while simplification. When you simplify between the two negative terms or two positive terms you have to do addition and give sign of bigger number while when you simplify between one negative term or the positive term you have to do subtraction and give sign of the bigger number to the resultant value.