Starting with this expression derived by Ellie82,
| 2r-3 | > -2/5
we recognize this as requiring two pairs of conditions.
First pair
The first condition is Ellie82's expression assuming that the absolute value operation does not change the sign of its argument. The second condition is that the argument is positive or zero, so the sign does not need to be changed.
(2r - 3) > -2/5, and (2r - 3) >= 0
Second pair
The first condition is Ellie82's expression assuming that the absolute value operation does change the sign of its argument. The second condition is that the argument is negative, so the sign needs to be changed.
-(2r - 3) > -2/5, and (2r - 3) < 0.
A casual examination of the first pair of conditions reveals the second of those to be the more restrictive. In other words,
2r - 3 >= 0
2r >= 3
r >= 3/2
The second pair of conditions can be rewritten as
-(2r - 3) > -2/5
(2r - 3) < 2/5 (multiplying both sides by -1. We recognize this as less restrictive than the second condition of this pair)
2r - 3 < 0 (second condition of second pair)
2r < 3 (add 3 to both sides)
r < 3/2 (divide both sides by 2)
When the first pair of conditions (yielding r >= 3/2 as a solution set) and the second pair of conditions (yielding r < 3/2 as a solution set) are combined, we find that
all values of r will satisfy the given inequality: 6+5 | 2r-3 | > 4.
| 2r-3 | > -2/5
we recognize this as requiring two pairs of conditions.
First pair
The first condition is Ellie82's expression assuming that the absolute value operation does not change the sign of its argument. The second condition is that the argument is positive or zero, so the sign does not need to be changed.
(2r - 3) > -2/5, and (2r - 3) >= 0
Second pair
The first condition is Ellie82's expression assuming that the absolute value operation does change the sign of its argument. The second condition is that the argument is negative, so the sign needs to be changed.
-(2r - 3) > -2/5, and (2r - 3) < 0.
A casual examination of the first pair of conditions reveals the second of those to be the more restrictive. In other words,
2r - 3 >= 0
2r >= 3
r >= 3/2
The second pair of conditions can be rewritten as
-(2r - 3) > -2/5
(2r - 3) < 2/5 (multiplying both sides by -1. We recognize this as less restrictive than the second condition of this pair)
2r - 3 < 0 (second condition of second pair)
2r < 3 (add 3 to both sides)
r < 3/2 (divide both sides by 2)
When the first pair of conditions (yielding r >= 3/2 as a solution set) and the second pair of conditions (yielding r < 3/2 as a solution set) are combined, we find that
all values of r will satisfy the given inequality: 6+5 | 2r-3 | > 4.
