Solve the inequality 25 - 2(n - 2) >= -8n + 6.

Step-by-step solution

1. 1

   Distribute the -2 across the parentheses. The minus sign belongs to the $2$, so both inner terms change: $-2\cdot n=-2n$ and $-2\cdot(-2)=+4$. Dropping that sign flip on the second term is the classic mistake here.

   $25-2n+4\ge -8n+6$

2. 2

   Combine the constants on the left. $25+4=29$, giving

   $29-2n\ge -8n+6$

3. 3

   Move the n terms to one side by adding 8n. Adding the same quantity to both sides never changes the direction of an inequality, and adding $8n$ (rather than subtracting $2n$) keeps the coefficient positive, which avoids a later flip:

   $29+6n\ge 6$

4. 4

   Isolate the n term. Subtract $29$ from both sides:

   $6n\ge -23$

5. 5

   Divide by 6 and explain why the sign holds. Because $6>0$, dividing preserves $\ge$:

   $n\ge -\frac{23}{6}$

   The fraction does not reduce, since $23$ is prime and does not divide $6$. As a decimal, $-\tfrac{23}{6}\approx-3.8\overline{3}$.

6. 6

   Check the boundary and one interior point. At $n=-\tfrac{23}{6}$: left $=25-2\left(-\tfrac{23}{6}-2\right)=25+\tfrac{70}{6}=\tfrac{220}{6}$ and right $=-8\left(-\tfrac{23}{6}\right)+6=\tfrac{184}{6}+\tfrac{36}{6}=\tfrac{220}{6}$ — equal, as expected at the boundary ✓. At $n=0$ (inside the solution set): $25-2(-2)=29\ge 6$ ✓, so the inequality really does point the way stated.