Solve the inequality |2x - 3| <= x - 1.

Step-by-step solution

1. 1

   Force the right-hand side to be non-negative first. The left side is an absolute value, so it is $\ge 0$. If $x-1<0$ the inequality is impossible, which gives a hard requirement:

   $x-1\ge 0\quad\Longrightarrow\quad x\ge 1$

   This gate is what makes a variable right-hand side different from a numeric one — skip it and you can pick up phantom solutions.

2. 2

   Replace the bars with a sandwich. For $b\ge 0$, $|A|\le b$ is equivalent to $-b\le A\le b$. With $A=2x-3$ and $b=x-1$:

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

3. 3

   Solve the left half. Expand and collect:

   $-x+1\le 2x-3\;\Longrightarrow\;4\le 3x\;\Longrightarrow\;x\ge\frac{4}{3}$

4. 4

   Solve the right half. Subtract $x$ and add $3$:

   $2x-3\le x-1\;\Longrightarrow\;x\le 2$

5. 5

   Intersect all three conditions. We need $x\ge 1$, $x\ge\frac43$ and $x\le 2$ simultaneously; the binding constraints are the last two:

   $\frac{4}{3}\le x\le 2$

   Check the endpoints: at $x=\tfrac43$, $\left|\tfrac83-3\right|=\tfrac13$ and $x-1=\tfrac13$ $\checkmark$; at $x=2$, $|1|=1$ and $x-1=1$ $\checkmark$. Outside, $x=3$ gives $|3|=3\not\le 2$ $\checkmark$.