Solve a system of linear inequalities without graphing, i.e. using pure algebra?
You can compare the right-hand sides as $$2x+1\geq\frac{x}{2}-1$$ or $$\frac{x}{2}-1\geq 2x+1$$
If $$2x+1\geq\frac{x}{2}-1$$ or $$x\geq-\frac{4}{3}$$ we obtain: $$\left\{(x,y)|x\geq-\frac{4}{3},y\geq2x+1\right\}\setminus\left\{-\frac{4}{3},-\frac{5}{3}\right\}.$$ If $$2x+1<\frac{x}{2}-1$$ or $$x<-\frac{4}{3}$$ we obtain: $$\left\{(x,y)|x<-\frac{4}{3},y>\frac{x}{2}-1\right\}.$$