Find the truth set of the inequality 2-3(x-2)<2-5x.

Answer 1

To solve the inequality $2-3(x-2)<2-5x$, we first simplify both sides.

Starting with the left side: $$2-3(x-2)=2-3x+6=8-3x$$

Now the inequality is: $$8-3x<2-5x$$

Next, we will isolate $x$. First, we add $5x$ to both sides: $$8-3x+5x<2$$ This simplifies to: $$8+2x<2$$

Now, subtract $8$ from both sides: $$2x<2-8$$ This simplifies to: $$2x<-6$$

Next, divide both sides by $2$: $$x<-3$$

Thus, the truth set of the inequality $2-3(x-2)<2-5x$ is: $$\boxed{{\displaystyle (-\mathrm{\infty },-3)}}$$