Question

What is the solution of $\left| {2x + 4} \right| = 8?$

Answer 1

In this problem we have the absolute value can be negative value or positive value. The absolute value is a measure of how far a number is from the origin with no consideration of it is direction. If we get minus value and the number inside the modulus when it comes out from the modulus there are positive values and negative values.

Complete step-by-step solution:
The given problem is $\left| {2x + 4} \right| = 8$
That is the minus value (or) positive value whatever it comes out the modulus it becomes a positive value.
That is $\left| { - 4} \right| = 4$ and $\left| 4 \right| = 4$
The absolute is a measure of how far a number of from the origin with no consideration of its direction.
Clear the absolute value term is $\left| {2x + 4} \right|$
For the negative we will use $- \left( {2x + 4} \right)$
For the positive we will use $+ \left( {2x + 4} \right)$
Solve the negative case
$- \left( {2x + 4} \right) = 8$
Multiply the term by the sign minus we have
$- 2x - 4 = 8$
Solve the equation we have
$- 2x = 8 + 4$
$\Rightarrow - 2x = 12$
$\Rightarrow x = \dfrac{{12}}{{ - 2}}$
$\Rightarrow x = - 6$
This is the solution for the negative case
Now, we have to solve the positive case
$2x + 4 = 8$
Rearrange this, we have
$2x = 8 - 4$
$2x = 4$
Solve this we have divide both sides by two
$\dfrac{{2x}}{2} = \dfrac{4}{2}$
$x = 2$
This is the solution for the positive case
Therefore the absolute solution for the given problem we get
$x = 2$
$\Rightarrow x = - 6$

Note: In mathematics absolute the magnitude of a real number without regard to its sign. Absolute value describes the distance from zero that a number is on the line, without considering direction. The absolute value of a number is never negative. When you see an absolute value is a problem or equation it means that whatever is inside the absolute value is always positive. Absolute values are often used with inequalities. That‘s the important thing to keep in mind; it's just like distance away from zero.