Solve the inequality x + 29 > 3x + 12.

Step-by-step solution

1. 1

   Send the variable to the side with the larger coefficient. Subtracting $x$ from both sides leaves $3x-x=2x$ on the right, a positive coefficient:

   $29>2x+12$

2. 2

   Move the constant. Subtract $12$ from both sides:

   $17>2x$

3. 3

   Divide by $2$ and keep the answer exact. $17$ is odd, so the boundary is genuinely fractional:

   $\frac{17}{2}>x\quad\Longleftrightarrow\quad x<\frac{17}{2}=8.5$

   Leaving it as $\tfrac{17}{2}$ is preferable to rounding — here $8.5$ happens to terminate, but the fraction is the exact form in general.

4. 4

   Check integers straddling $8.5$. At $x=8$: left $=37$, right $=36$, and $37>36$ $\checkmark$. At $x=9$: left $=38$, right $=39$, false $\checkmark$. At $x=8.5$ both sides equal $37.5$, so the strict inequality excludes the boundary. Solution set: $\left(-\infty,\tfrac{17}{2}\right)$.