Solve for a.

Question

5a<a+20≤8+5a
Write your answer as a compound inequality with integers.

Answer 1

To solve the compound inequality \(5a < a + 20 \leq 8 + 5a\), we will break it down into two parts and solve each part separately.

Step 1: Solve the first part \(5a < a + 20\)

1. Subtract \(a\) from both sides: \[ 5a - a < 20 \] This simplifies to: \[ 4a < 20 \]
2. Divide both sides by 4: \[ a < 5 \]

Step 2: Solve the second part \(a + 20 \leq 8 + 5a\)

1. Subtract \(a\) from both sides: \[ 20 \leq 8 + 5a - a \] This simplifies to: \[ 20 \leq 8 + 4a \]
2. Subtract 8 from both sides: \[ 12 \leq 4a \]
3. Divide both sides by 4: \[ 3 \leq a \quad \text{or} \quad a \geq 3 \]

Final Step: Combine the results

From the two parts, we have:

1. \(a < 5\)
2. \(a \geq 3\)

Combining these gives us the compound inequality: \[ 3 \leq a < 5 \]

Conclusion: The final solution is: \[ \boxed{[3, 5)} \]