Solve the absolute value equation |3x - 5| = 5.

Step-by-step solution

1. 1

   Check that the right-hand side is positive. An absolute value is a distance, so it is never negative. Here the right side is $5>0$, which means solutions exist and there will be exactly two of them — one for each of the two points that sit $5$ units from zero.

2. 2

   Turn the one equation into two. The statement $|A|=5$ says $A$ is $5$ units from $0$, so $A=5$ or $A=-5$. With $A=3x-5$:

   $3x-5=5\qquad\text{or}\qquad 3x-5=-5$

   Nothing has been lost or gained — the two cases together are exactly equivalent to the original equation.

3. 3

   Solve the positive case. Add $5$ to both sides and divide by $3$:

   $3x=10\quad\Longrightarrow\quad x=\frac{10}{3}$

4. 4

   Solve the negative case. Add $5$ to both sides again; this time the right side collapses to zero:

   $3x-5=-5\quad\Longrightarrow\quad 3x=0\quad\Longrightarrow\quad x=0$

5. 5

   Substitute both answers back. Checking is cheap and catches sign slips:

   $\left|3\cdot\tfrac{10}{3}-5\right|=|10-5|=5\qquad\checkmark$
   $|3\cdot 0-5|=|-5|=5\qquad\checkmark$

   Both values satisfy the original equation, so the solution set is $\left\{0,\tfrac{10}{3}\right\}$.