Question

Solve the given inequality for real $x$:

$3(x-1)\le 2(x-3)$

Answer 1

Solve the inequality for real $x$:

$$\begin{gathered} 3(x–1)\le 2(x–3) \\ \Rightarrow 3x–3\le 2x–6 \end{gathered}$$
Adding $3$ on both sides, we obtain
$$\begin{gathered} 3x–3+3\le 2x–6+3 \\ \Rightarrow 3x\le 2x–3 \end{gathered}$$
Subtracting $2x$ from both sides,
$$\begin{gathered} 3x–2x\le 2x–3–2x \\ \Rightarrow x\le -3 \end{gathered}$$
That means, all the real numbers which are less than or equal to $-3$ are the solutions for the given inequality.
Therefore, $\mathit{x}\mathbf{}\mathbf{\in }\mathbf{}\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{}\mathbf{-}\mathbf{3}\mathbf{]}$ is the solution of given inequality.