Solve the inequality (2 - x - x)/3 < (1/3)(2 - 5x).

Step-by-step solution

1. 1

   Tidy the numerator first. The left numerator contains two like terms: $2-x-x=2-2x$, so the inequality is $\frac{2-2x}{3}<\frac13(2-5x).$

2. 2

   Clear the denominators, and note the sign rule. Both sides carry a factor $\frac13$. Multiplying an inequality by a positive number preserves its direction, and $3>0$, so the sign stays as $<$: $2-2x<2-5x .$ This is the step where multiplying by a negative would have required flipping the inequality.

3. 3

   Move the variable terms to one side. Adding $5x$ to both sides gives $2+3x<2 .$

4. 4

   Isolate x. Subtracting $2$ leaves $3x<0$, and dividing by the positive number $3$ gives $x<0 .$

5. 5

   Check with a test value from each side. At $x=-1$: left $=\frac{2+2}{3}=\frac43$, right $=\frac13(2+5)=\frac73$, and $\frac43<\frac73$ is true. At $x=1$: left $=0$, right $=\frac13(-3)=-1$, and $0<-1$ is false. So the solution set is exactly $x<0$, i.e. $(-\infty,0)$.