Question

How do you solve$\left| {2x - 5} \right| + 3 = 12$?

Answer 1

In the given question the equation is an absolute value equation. Absolute value means it shows how far a number is from zero.
The absolute value equation always has two answers, one if positive and one is negative.
We will find the value of $x$using both positive and negative values of the absolute equation.

Complete step by step solution:
The given equation is an absolute value equation, that is
$\left| {2x - 5} \right| + 3 = 12$
But in absolute value equation we know that
$\left| x \right| = + x$ And $- x$
Now we will use the above concept and we will break $\left| {2x - 5} \right|$ into two equations. Now the resulted equations will be
$2x - 5 + 3 = 12$……………….. (i)
$- \left( {2x - 5} \right) + 3 = 12$…………… (ii)
Now we will solve both the equation to get the value of $x$
We will solve the equation (i)
$2x - 5 + 3 = 12$
Now we will simplify the left side of the equation and the resulted equation will be
$\Rightarrow 2x - 2 = 12$
Now we will separate the like terms in the above equation,
$\Rightarrow 2x = 12 + 2$
$\Rightarrow 2x = 14$
$\Rightarrow x = \dfrac{{14}}{2}$
$\Rightarrow x = 7$
Now we will take the equation (ii) and solve it to get the value of $x$
$- \left( {2x - 5} \right) + 3 = 12$
Now we will open the bracket and we get,
$\Rightarrow - 2x + 8 = 12$
Now we will separate the like terms in the above equation,
$\Rightarrow - 2x = 12 - 8$
$\Rightarrow - 2x = 4$
$\Rightarrow x = - \dfrac{4}{2}$
$\Rightarrow x = - 2$
So as we calculated above the solution of the given equation are
$x = 7, - 2$

Note: We know that absolute value equations always have two values one is positive and one is negative. Always find the value of $x$using both positive and negative values of the absolute equation.