[tex]1.\\d+7>\dfrac{7d+3}{3}\qquad\text{multiply both sides by 3}\\\\3d+21>7d+3\qquad\text{subtract 21 from both sides}\\\\3d>7d-18\qquad\text{subtract 7d from both sides}\\\\-4d>-18\qquad\text{change the signs}\\\\4d<18\qquad\text{divide both sides by 4}\\\\\boxed{d<4.5}[/tex]
[tex]2.\\\dfrac{a-6}{3}>a+2\qquad\text{multiply both sides by 3}\\\\a-6>3a+6\qquad\text{add 6 to both sides}\\\\a>3a+12\qquad\text{subtract 3a from both sides}\\\\-2a>12\qquad\text{change the signs}\\\\2a<-12\qquad\text{divide both sides by 2}\\\\\boxed{a<-6}[/tex]
[tex]3.\\\dfrac{d}{2}<\dfrac{3d}{2}+6\qquad\text{multiply both sides by 2}\\\\d<3d+12\qquad\text{subtract 3d from both sides}\\\\-2d<12\qquad\text{change the signs}\\\\2d>-12\qquad\text{divide both sides by 2}\\\\\boxed{d>-6}[/tex]
[tex]4.\\36-2n<\dfrac{1}{3}n+15\qquad\text{subtract 36 from both sides}\\\\-2n<\dfrac{1}{3}n-21\qquad\text{multiply both sides by 3}\\\\-6n<n-63\qquad\text{subtract n from both sides}\\\\-7n<-63\qquad\text{change the signs}\\\\7n>63\qquad\text{divide both sides by 7}\\\\\boxed{n>9}[/tex]
