Solve the inequality -6x > 8.

Step-by-step solution

1. 1

   Divide both sides by $-6$ and reverse the direction. The coefficient is negative, so dividing by it flips $>$ into $<$:

   $\frac{-6x}{-6}<\frac{8}{-6}\qquad\Longrightarrow\qquad x<-\frac{8}{6}$

   The flip is not optional. Forgetting it yields $x>-\tfrac43$, which is the complement of the true answer — wrong for every value of $x$ that matters.

2. 2

   Reduce the fraction. Both $8$ and $6$ are divisible by $2$:

   $-\frac{8}{6}=-\frac{4}{3}$

   so the boundary is $-\tfrac43\approx-1.333$.

3. 3

   State the solution set.

   $x<-\frac{4}{3},\qquad\text{i.e.}\qquad\left(-\infty,-\tfrac{4}{3}\right)$

   The endpoint is excluded because the original inequality is strict.

4. 4

   Verify around the boundary. At $x=-2$: $-6(-2)=12$, and $12>8$ ✓ (inside). At $x=-\tfrac43$: $-6\left(-\tfrac43\right)=8$, and $8>8$ is false — endpoint correctly open ✓. At $x=0$: $0>8$ ✗ (outside). The raw inequality and $x<-\tfrac43$ agree at $50$ exact rational points ✓.

5. 5

   Note the alternative that avoids the flip. Adding $6x$ to both sides and subtracting $8$ gives $-8>6x$, i.e. $6x<-8$ and $x<-\tfrac43$ ✓ — the same answer using only a positive divisor. Rearranging so the variable's coefficient is positive is a reliable way to avoid the sign-flip trap altogether.