The set of all values of x such that 3 (3 – 2x) ≤ 15 is:

A. x ≤ 1

B. x ≥ 1

C. x ≤ -1

D. x ≤ -2

E. None of the above

does the sign change? not sure

You need to know why the sign would change....

\(\displaystyle 5>4\) but \(\displaystyle -5<-4\)

If it was temperature, the temperature dropping from 5 would go to 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6.....

How would we go from \(\displaystyle 5>4\) to \(\displaystyle -5<-4\)

This would happen if we multiply or divide both sides by the same negative value.

Changing signs is the mathematical operation of multiplying or dividing by \(\displaystyle -1\)

It does not matter if we have an equality... \(\displaystyle x=y\ \Rightarrow\ -x=-y\)

but \(\displaystyle x>y\ \Rightarrow\ -x<-y\)

Therefore,

\(\displaystyle 3(3-2x)\ \le\ 15\ \Rightarrow\ 3-2x\ \le\ 5\ \Rightarrow\ 3\ \le\ 5+2x\)

\(\displaystyle 3-5\ \le\ 2x\)

\(\displaystyle -2\ \le\ 2x\)

\(\displaystyle -1\ \le\ x\ \Rightarrow\ x\ \ge\ -1\)

You can reverse the inequality also to bring you to the same answer.

Notice above that we did not do any multiplying or dividing by negative numbers,

hence no reversal was ever needed.

\(\displaystyle 3(3-2x)\ \le\ 15\)

\(\displaystyle 3-2x\ \le\ 5\)

Now, multiply (or divide) both sides by \(\displaystyle -1\) to get the co-efficient of x positive.

However, you can add 2x to both sides to also get the co-efficient of x positive.

Getting the co-efficient of x positive is necessary to find it's boundary.

\(\displaystyle 2x-3\ ?\ -5\)

What goes between ?