Final answer:
To solve the given inequalities, we need to find the values of 'r' that satisfy both of them. For the first inequality 1/2r + 3 < 7/4, we subtract 3 from both sides and simplify to get r < -5/2. For the second inequality -r + 3/4 ≤ 3/8, we subtract 3/4 from both sides and simplify to get r ≥ 3/8. Therefore, the values of 'r' that satisfy both inequalities are -5/2 < r ≥ 3/8.
Step-by-step explanation:
To solve the given inequalities, we need to find the values of 'r' that satisfy both of the inequalities. Let's solve them one at a time:
For the first inequality, 1/2r + 3 < 7/4, subtract 3 from both sides: 1/2r < 7/4 - 3. Simplifying, we get 1/2r < 7/4 - 12/4, which becomes 1/2r < -5/4. Now, multiply both sides by 2 to get rid of the fraction: r < -5/4 * 2. Multiplying further, we find r < -10/4, which simplifies to r < -5/2.
For the second inequality, -r + 3/4 ≤ 3/8, subtract 3/4 from both sides: -r ≤ 3/8 - 3/4. Simplifying, we get -r ≤ 3/8 - 6/8, which becomes -r ≤ -3/8. Now, multiply both sides by -1 and reverse the inequality sign: r ≥ 3/8.
Therefore, the values of 'r' that satisfy both inequalities are -5/2 < r ≥ 3/8.