Given composite inequality : -4+r < 4r+8 and -4r+9 ≤ 1-5r
Let us solve first and second part of the given inequalities one by one.
Let us solve -4+r < 4r+8 first.
We need to get rid -4 from left side. In order to get rid -4, we need to add 4 on both sides.
On adding 4 both sides, we get
-4+4+r < 4r+8+4
On simplifying, we get
r < 4r +12.
Subtracting 4r from both sides, we get
r-4r < 4r -4r +12
-3r < 12.
Dividing both sides by -3, we get
r > -4.
Note : On dividing by a negative number, the inequality sign get flip.
In the above problem < sign became >.
Let us solve second part -4r+9 ≤ 1-5r now.
-4r+9 ≤ 1-5r
Subtracting 9 from both sides, we get
-4r+9-9 ≤ 1-9-5r
-4r ≤ -8 -5r.
Adding 5r from both sides, we get
-4r+5r ≤ -8 -5r+5r
r ≤ -8.
Therefore, final answer is
r > -4 and r ≤ -8.