Question

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

Answer 1

Hint: 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.