Answer:
the solution to the inequality 1/2 + r/-4 ≥ 5/6 is r ≤ -4/3.
Explanation:
To solve the inequality 1/2 + r/-4 ≥ 5/6, we need to isolate the variable r.
1. First, let's simplify the expression on the left side of the inequality:
1/2 + r/-4 can be rewritten as 1/2 - r/4.
2. Next, let's find a common denominator for 1/2 and -r/4. The common denominator is 4.
1/2 - r/4 becomes 2/4 - r/4.
3. Now, we can combine the fractions:
2/4 - r/4 = (2 - r)/4.
4. The inequality becomes:
(2 - r)/4 ≥ 5/6.
5. To eliminate the denominator, we can multiply both sides of the inequality by 4:
4 * (2 - r)/4 ≥ 4 * 5/6.
This simplifies to:
2 - r ≥ 20/6.
6. Let's further simplify the right side:
20/6 can be simplified to 10/3.
So the inequality becomes:
2 - r ≥ 10/3.
7. To isolate the variable r, we can subtract 2 from both sides of the inequality:
2 - r - 2 ≥ 10/3 - 2.
This simplifies to:
-r ≥ 4/3.
8. Finally, to solve for r, we need to multiply both sides of the inequality by -1. Since we are multiplying by a negative number, we need to reverse the direction of the inequality:
-r * (-1) ≤ (4/3) * (-1).
This simplifies to:
r ≤ -4/3.
Therefore, the solution to the inequality 1/2 + r/-4 ≥ 5/6 is r ≤ -4/3.