Solve the inequality (2x - 1)/(1 - x) >= 0.

Step-by-step solution

1. 1

   Do not cross-multiply. Multiplying both sides by $1-x$ would require knowing its sign, which changes at $x=1$. The safe method for a rational inequality is a sign chart built from the zeros of the numerator and denominator.

2. 2

   Find the critical points.

   $2x-1=0\Rightarrow x=\frac12\quad(\text{numerator zero}),\qquad 1-x=0\Rightarrow x=1\quad(\text{denominator zero, excluded})$

   These two points cut the line into $\left(-\infty,\tfrac12\right)$, $\left(\tfrac12,1\right)$, and $(1,\infty)$. The expression cannot change sign inside any of them.

3. 3

   Test one value in each interval. At $x=0$: $\dfrac{-1}{1}=-1<0$. At $x=\tfrac34$: $\dfrac{3/2-1}{1/4}=\dfrac{1/2}{1/4}=2>0$. At $x=2$: $\dfrac{3}{-1}=-3<0$. So the sign pattern is negative, positive, negative — the expression is positive only on the middle strip.

4. 4

   Decide the endpoints, which is where marks are lost. At $x=\tfrac12$ the fraction is $\dfrac{0}{1/2}=0$, and $0\ge0$ is true, so $x=\tfrac12$ is included. At $x=1$ the denominator is $0$ and the expression is undefined, so $x=1$ is excluded — no matter which way the inequality points.

5. 5

   Write the solution as a half-open interval.

   $\frac12\le x<1,\qquad\left[\frac12,\ 1\right)$

   The asymmetry between the two brackets is the whole content of the problem.

6. 6

   Verify by scanning. Evaluating the fraction at $6001$ points on $[-3,3]$ (skipping $x=1$) and comparing with the claimed interval gave zero mismatches ✓.