Solve the inequality 25 - 3(n - 2) >= -8n + 6.

Step-by-step solution

1. 1

   Distribute the -3, watching the second sign. $-3\cdot n=-3n$ and $-3\cdot(-2)=+6$, so the parentheses open to

   $25-3n+6\ge -8n+6$

2. 2

   Combine the constants on the left. $25+6=31$:

   $31-3n\ge -8n+6$

3. 3

   Add 8n to both sides. This gathers the variable on the left with a positive coefficient, so no direction flip will be needed later:

   $31+5n\ge 6$

4. 4

   Subtract 31 to isolate 5n.

   $5n\ge -25$

5. 5

   Divide by the positive 5. Since $5>0$ the $\ge$ survives untouched, and $-25$ is divisible by $5$, so the bound is a clean integer:

   $n\ge -5$

   This is the structural difference from the $-2$ version of the same problem, where the bound comes out as $-\tfrac{23}{6}$.

6. 6

   Verify the boundary and both sides of it. At $n=-5$: left $=25-3(-7)=46$ and right $=40+6=46$ — equality holds ✓. At $n=0$ (inside): $25+6=31\ge6$ ✓. At $n=-6$ (outside): left $=25-3(-8)=49$, right $=48+6=54$, and $49\ge54$ is false ✓, confirming the solution set is exactly $[-5,\infty)$.