Solve the inequality (1 + x)/3 < x - 1.

Step-by-step solution

1. 1

   Clear the single denominator. Multiplying both sides by $3$ removes the fraction, and because $3>0$ the inequality sign does not flip:

   $1+x<3(x-1)$

   Every term on the right must be multiplied, which is why the bracket is needed.

2. 2

   Expand the right-hand side.

   $1+x<3x-3$

3. 3

   Collect the variable terms on one side. Subtract $x$ from both sides:

   $1<2x-3$

   Moving $x$ to the right (rather than $3x$ to the left) keeps the coefficient of $x$ positive, so no sign flip will be needed at the end.

4. 4

   Isolate the $x$ term. Add $3$:

   $4<2x$

5. 5

   Divide by 2 and check. Dividing by the positive $2$ gives $2<x$, i.e.

   $x>2,\qquad x\in(2,\infty)$

   Check the boundary $x=2$: both sides equal $1$, so the strict inequality fails and $2$ is excluded. Check $x=3$: the left side is $\frac43\approx 1.33$ and the right side is $2$, so $1.33<2$ holds ✓.