Answer:
k = -3
Explanation:
You want the value of k that makes k(4r-5)>-12r-9 true for all real numbers.
Tautology
The given inequality will be true for all values of r when the coefficient of r in the rearranged expression is zero.
We want to rearrange the expression to the form ...
ar +b > 0
We can do this by subtracting (-12r-9) from both sides.
k(4r -5) -(-12r -9) > 0
(4k+12)r -5k +9 > 0
Making the coefficient of r be zero, we find k to be ...
4k +12 = 0 ⇒ k = -12/4 = -3
Check
Using this value of k in the expression gives ...
(4(-3) +12)r -5(-3) +9 > 0
24 > 0 . . . . . . . . . . simplify; true for all values of r
The value of k for which the solution is "all real numbers" is -3.