Number line arrangement

On a number line, consider the points corresponding to the values a, 2a, and 3a + 1 arranged in all possible orders. For each arrangement, decide whether it is possible. If yes, give a specific example; if not, explain why.

Final Answer:

a₁ =  1
a₂ =  -5
a₃ =  -0.75
a₄ =  -0.25

Step-by-step explanation:

$$\begin{gathered} 2a=3a+1=>a=-1 \\ 2a=a=>a=0 \\ a=3a+1=>a=-0.5 \\ D=\{-1,-0.5,0\} \\ a,2a,3a+1 \\ {a}_1=1 \\ {a}_1=1<2<4=1 \end{gathered}$$

$$\begin{gathered} 3a+1,2a,a \\ {a}_2=-5 \\ {a}_2=-14<-10<-5=-5 \end{gathered}$$

$$\begin{gathered} 2a,3a+1,a \\ {a}_3=-0.75 \\ {a}_3=-1.5<-1.25<-0.75=-0.75 \end{gathered}$$

$$\begin{gathered} 2a,a,3a+1 \\ {a}_4=-0.25 \\ {a}_4=-0.5<-0.25<0.25=-0.25 \end{gathered}$$

Try another example