Example 3.6.7

Solve for \(x\) in the inequality \(-5x+1\le -5x\text{.}\)

To solve for \(x\text{,}\) we will first add \(5x\) to each side to get all terms containing \(x\) on one side:

\begin{align*} -5x+1\amp\le -5x\\ -5x+1\addright{5x}\amp\le -5x\addright{5x}\\ 1\amp\le 0 \end{align*}

Once more, the variable \(x\) is absent. So we can ask ourselves, “For which values of \(x\) is \(1\le 0\) true?” The answer is none, and so there is no solution to this inequality. We can write the solution set using \(\emptyset\text{.}\)