Find the root of the equation 1/(6x + 8) = 5.

Step-by-step solution

1. 1

   Note the excluded value. The denominator is zero when $6x + 8 = 0$, i.e. $x = -\tfrac43$. Any candidate root equal to that must be discarded.

2. 2

   Take reciprocals of both sides. Since neither side is zero, inverting is valid and is quicker than cross-multiplying:

   $6x + 8 = \frac{1}{5} = 0.2$

3. 3

   Solve the resulting linear equation.

   $6x = 0.2 - 8 = -7.8 \quad\Longrightarrow\quad x = \frac{-7.8}{6} = -1.3$

   As a fraction, $x = -\tfrac{39}{30} = -\tfrac{13}{10}$.

4. 4

   Confirm the root is admissible. $-1.3 \ne -\tfrac43 \approx -1.3\overline{3}$ — close, but not equal, so the root is valid.

5. 5

   Verify in the original equation. $6(-1.3) + 8 = -7.8 + 8 = 0.2$, and $\tfrac{1}{0.2} = 5$. Correct.