Question

How do you solve $\left| {2x\, + \,\left. 6 \right|} \right.\, - \,4\, = \,20$ ?

Answer 1

Hint: The absolute value function can take any negative and positive value and transform it to a positive value. Here, we take $4$ to the right-hand side of the equation and then remove the absolute value function by taking both the positive and negative values the function can take.

Formula used:
When removing the sign of the absolute value function, it can be written as
 $x\, = \,a$ or
 $x\, = \, - a$

Complete step-by-step solution:
The above equation can be written below:
 $\left| {2x\, + \,\left. 6 \right|} \right.\, - \,4\, = \,20$
Transposing $4$ to the right-hand side of the equation
$\Rightarrow$ $\left| {2\,x + \left. 6 \right|} \right.\, = \,24$
Now removing the absolute value sign.
$\Rightarrow$ $2\,x + 6 = 24\,\,or\,\,2x + 6\, = \, - 24$
Subtracting $6$ from both the sides of both the equations.
$\Rightarrow$ $2\,x + 6 - 6 = 24 - 6\,\,or\,\,2x + 6 - 6\, = \, - 24 - 6$
Writing the final equation.
$\Rightarrow$ $2\,x = 18\,\,or\,\,2x\, = \, - 30$
Dividing the whole equation by $2$
$\Rightarrow$\[\dfrac{{2\,x}}{2} = \dfrac{{18}}{2}\,\,or\,\,\dfrac{{2x}}{2}\, = \, - \dfrac{{30}}{2}\]
Writing the equation after division.
$\Rightarrow$ $\,x = 9\,or\,\,x\, = \, - 15$

Thus, we obtain two values of $x$ . Therefore $x$ can take two values $9\,and\, - 15$ .

Note: Absolute value of a number will always yield a positive number. Therefore, there will be two answers pertaining to the question, one has to be positive the other negative.
The meaning of Absolute value is the distance of a point from zero. Since distance cannot be negative hence, it always obtains a positive value. There can be a case when the value of the absolute value function is zero. For example, $\left| x \right|\, = \, - 1$ has no solution. There can be a case when the absolute value function has one solution.
 For example, $\left| x \right|\, = 0$ has only one solution i.e. $0$ .