Solve the inequality 3x + 9 > 7x + 6.

Step-by-step solution

1. 1

   Subtract the smaller $x$ term to keep the coefficient positive. Removing $3x$ (rather than $7x$) leaves a positive $4x$ and avoids dividing by a negative later:

   $9>4x+6$

2. 2

   Move the constant across. Subtract $6$ from both sides:

   $3>4x$

3. 3

   Divide by 4. The divisor is positive, so the direction is unchanged:

   $\frac{3}{4}>x$

4. 4

   Turn the statement around. Swapping the sides of an inequality reverses the symbol, so $\tfrac34>x$ is the same as

   $x<\frac{3}{4}$

   In interval notation $\left(-\infty,\tfrac34\right)$, with the endpoint excluded because the relation is strict.

5. 5

   Verify around the boundary. At $x=0$: left $9$, right $6$, and $9>6$ ✓ (inside). At $x=1$: left $12$, right $13$ ✗ (outside). At $x=\tfrac34$: left $9+\tfrac94=\tfrac{45}{4}$ and right $\tfrac{21}{4}+6=\tfrac{45}{4}$ — exactly equal, confirming the open endpoint. The raw inequality and $x<\tfrac34$ agree at $80$ exact rational points ✓.