Question

What do you do when you have absolute values on both sides of the equations?

Answer 1

Hint: When we have absolute values on both the sides of the equation, then we will consider both the possibilities of getting positive values and the negative values. Here we will take one example to understand it.

Complete step-by-step solutions:
Take the expression: $\left| {7 + 2x} \right| = \left| 9 \right|$
Now remove mode and get the two equations- on removing the mode we have the possibilities of getting positive values and the negative values like
$\Rightarrow (7 + 2x) = - 9$
$\Rightarrow (7 + 2x) = + 9$
Now solve equations (A) and (B) one by one-
$\Rightarrow (7 + 2x) = - 9$
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.
$\Rightarrow 2x = - 9 - 7$
Simplify the above equation
$\Rightarrow 2x = - 16$
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
$
 \Rightarrow x = - \frac{{16}}{2} \\
\Rightarrow x = ( - 8)\;{\text{ }} \ldots .{\text{ }}\left( C \right) \\
$
$\Rightarrow (7 + 2x) = 9$
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 term becomes negative and negative term becomes positive.
$\Rightarrow 2x = 9 - 7$
Simplify the above equation
$\Rightarrow 2x = 2$
Make “X” the subject
$ \Rightarrow x = \frac{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 $x = 1$

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.