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

Step-by-step solution

1. 1

   Record the excluded value first. The denominator vanishes at $x = -6$, so whatever root appears, it must not equal $-6$. Noting this before clearing the fraction is what makes the final check meaningful.

2. 2

   Multiply both sides by (x + 6).

   $8 = -5(x+6)$

   This is legitimate exactly because $x \ne -6$ makes the multiplier non-zero.

3. 3

   Expand and collect.

   $8 = -5x - 30 \quad\Longrightarrow\quad 5x = -38 \quad\Longrightarrow\quad x = -\frac{38}{5} = -7.6$

4. 4

   Check the root against the exclusion. $-7.6 \ne -6$, so the root is admissible.

5. 5

   Verify in the original equation. $x + 6 = -7.6 + 6 = -1.6$, and $\tfrac{8}{-1.6} = -5$. The sign works out because a positive numerator over a negative denominator is negative, as the right-hand side demands.