- The first inequality, 1/2 * r + 3 < 7/4, has the solution r < -1/2.
- The second inequality, -r + 3/4 <= 3/8, has the solution r >= 3/4.
Therefore, the only value of r that satisfies both inequalities is r >= 3/4.
We analyzed two inequalities involving r. By isolating r, manipulating equations, and simplifying, we found that the first inequality is satisfied when r is less than -1/2. The second inequality is satisfied when r is greater than or equal to 3/4. The only value of r that fulfills both conditions is r >= 3/4.
Solving the Inequalities:
1/2 * r + 3 < 7/4
1. Isolate r: Subtract 3 from both sides:
1/2 * r < 7/4 - 3
2. Multiply both sides by 2:
r < 7/2 - 3
3. Combine like terms:
r < -1/2
-r + 3/4 <= 3/8
1. Add r to both sides:
3/4 <= r + 3/4
2. Simplify:
3/4 <= r