For which values of x is the fraction (2 − 3x)/(x(x − 1)) positive?

Step-by-step solution

1. 1

   Find every point where the sign can change. These are the zeros of the numerator and of the denominator:

   $2-3x=0\Rightarrow x=\tfrac23;\qquad x=0;\qquad x-1=0\Rightarrow x=1$

   They split the line into four intervals: $(-\infty,0)$, $\left(0,\tfrac23\right)$, $\left(\tfrac23,1\right)$ and $(1,\infty)$.

2. 2

   Test the interval x < 0 (take $x=-1$). Numerator $2-3(-1)=5>0$; denominator $(-1)(-2)=2>0$ (negative times negative). The fraction is positive ✓.

3. 3

   Test 0 < x < 2/3 (take $x=0.5$). Numerator $2-1.5=0.5>0$; denominator $(0.5)(-0.5)=-0.25<0$. The fraction is negative ✗.

4. 4

   Test 2/3 < x < 1 (take $x=0.8$). Numerator $2-2.4=-0.4<0$; denominator $(0.8)(-0.2)=-0.16<0$. Negative over negative is positive ✓.

5. 5

   Test x > 1 (take $x=2$). Numerator $2-6=-4<0$; denominator $(2)(1)=2>0$. The fraction is negative ✗.

6. 6

   Collect the positive intervals. All three critical points are excluded — $x=\tfrac23$ makes the fraction zero (not positive) and $x=0,1$ make it undefined:

   $\boxed{\{x<0\}\cup\left\{\tfrac23<x<1\right\}}$

7. 7

   Cross-check with the alternation rule. All three factors are simple (odd multiplicity), so the sign must alternate at each critical point. Reading left to right the pattern is $+,-,+,-$ — exactly what the four test values produced, which confirms no interval was mis-signed.