Solve the inequality 8g - 6(g + 1) < 4(2g - 9).

Step-by-step solution

1. 1

   Distribute on both sides.

   $8g-6g-6<8g-36$

   On the left, $-6(g+1)$ gives $-6g-6$; on the right, $4(2g-9)$ gives $8g-36$. Both terms inside each bracket must be multiplied.

2. 2

   Combine like terms on the left.

   $8g-6g=2g\quad\Rightarrow\quad 2g-6<8g-36$

3. 3

   Move the $g$ terms to the larger-coefficient side. Subtracting $2g$ (rather than $8g$) leaves a positive coefficient:

   $-6<6g-36$

   This choice is what lets us finish without ever multiplying or dividing by a negative — so the inequality sign never has to flip.

4. 4

   Collect the constants. Add $36$ to both sides:

   $30<6g$

5. 5

   Divide by the positive 6, then check. Since $6>0$ the direction is preserved:

   $5<g,\qquad\text{i.e.}\qquad g>5$

   Test $g=6$: left $=48-42=6$, right $=4(3)=12$, and $6<12$ ✓. Test $g=5$: left $=40-36=4$, right $=4(1)=4$, and $4<4$ is false, so $5$ is correctly excluded. The solution set is $(5,\infty)$.