9|r-2|-10<-73

r<-5
r<9

When I type it into a calculator online it says it is no solution but I am confused on why that is

First thing I did was add 10 to the other side so I got -63

Then I did 9|r-2|<-63 or 9|r-2|<63
Next I divide 9 and got -7 on one and 7 on the other and then added 2 so I got -5 and 9.

9|r-2|-10<-73

9|r-2| < -63
|r - 2| < -7

BUT, the absolute value of anything has to be ≥ 0, so it certainly cannot be ≤ -7

Your inequation has no solution.

Just try some numbers in your solution, they don't work.
e.g. for r < 9 , let r = 4
LS = 9|4-2) - 10 = 8
RS = -73
is 8 < -73 ?? , not when I went to school.

To solve the inequality 9|r-2|-10 < -73, we'll break it down into two cases:

Case 1: r - 2 >= 0
In this case, we can remove the absolute value and rewrite the inequality as:
9(r - 2) - 10 < -73
9r - 18 - 10 < -73
9r - 28 < -73
9r < -45
r < -5

So, for r - 2 >= 0, the solution is r < -5.

Case 2: r - 2 < 0
In this case, the inequality inside the absolute value becomes the negation of the original inequality. So we rewrite it as:
-9(r - 2) - 10 < -73
-9r + 18 - 10 < -73
-9r + 8 < -73
-9r < -81
r > 9

So, for r - 2 < 0, the solution is r > 9.

Combining both cases, we find that the solution to the original inequality 9|r-2|-10 < -73 is:
r < -5 or r > 9.