Question

I really need help with this problem!!

Answer 1

1.

`4<7 \underbrace{\implies}_(* 3) 12<21 \underbrace{\implies}_(* 2) 24<42 \underbrace{\implies}_(* 4) 96<168 \underbrace{\implies}_(* 9) 864<1512`

2.

`11>-2 \underbrace{\implies}_(+3) 14>1 \underbrace{\implies}_(+3) 17>4\underbrace{\implies}_(+(-4)) 13>0`

3.

`-2<-2\underbrace{\implies}_(-6)-8<-8\underbrace{\implies}_(-8)-16<-16\underbrace{\implies}_(-2)-18<-18`

4.

`-4<8\underbrace{\implies}_(/ (-4))1<-2 \underbrace{\implies}_(/ (-2)) -(1)/(2)<1`

Comments:

We can see that as long as we add/subtract any value to both sides nothing happens: true statements remain true and false statements remain false. But if we multiply or divide by negative values, false statements become true and true statements become false. That why, when you multiply or divide an inequality by a negative number, you should also flip the inequality sign. For example, if we start with the true inequality

`-2<1`

If we multiply both sides by -2 we have to flip the sign as well:

`4>-2`

And the inequality will remain true.