Solve the inequality x - 1/3 > x + 4.

Step-by-step solution

1. 1

   Collect the variable terms as usual. Both sides carry exactly one $x$ with coefficient $1$, so subtract $x$ from both sides:

   $x-\frac{1}{3}-x>x+4-x$

2. 2

   Notice the variable disappears completely. The $x$ terms cancel on both sides, leaving a statement with no variable in it at all:

   $-\frac{1}{3}>4$

   This is not a dead end — it is the answer in disguise. When the variable vanishes, the surviving numeric claim decides the whole problem at once.

3. 3

   Judge the surviving statement. Is $-\tfrac13$ greater than $4$? No: $-\tfrac13\approx-0.333$ is negative while $4$ is positive, so the claim is false. Since it does not depend on $x$, it is false for every $x$ simultaneously.

4. 4

   Conclude with the empty set. No real number satisfies the inequality, so

   $\text{solution set}=\varnothing$

   Testing $200$ exact rational values of $x$ produced no solution ✓ — as expected, since the two sides differ by a constant.

5. 5

   See why geometrically. The lines $y=x-\tfrac13$ and $y=x+4$ have the same slope $1$, so they are parallel and never cross. The left line sits a constant $4+\tfrac13=\tfrac{13}{3}$ below the right one, so it can never be above it. Had the inequality been reversed, $x-\tfrac13<x+4$, the leftover statement $-\tfrac13<4$ would be true and the solution set would be all real numbers.