1< y+10/3 ≤5

Question

Write your answer as a compound inequality with integers.

Answer 1

To solve the inequality \( 1 < \frac{y + 10}{3} \leq 5 \), we'll break it into two parts and solve each part separately.

Step 1: Solve the left part of the inequality

1. Start with the left side: \[ 1 < \frac{y + 10}{3} \]
2. Multiply both sides by 3: \[ 3 < y + 10 \]
3. Subtract 10 from both sides: \[ 3 - 10 < y \] \[ -7 < y \quad \text{or} \quad y > -7 \]

Step 2: Solve the right part of the inequality

1. Now, solve the right side: \[ \frac{y + 10}{3} \leq 5 \]
2. Multiply both sides by 3: \[ y + 10 \leq 15 \]
3. Subtract 10 from both sides: \[ y \leq 15 - 10 \] \[ y \leq 5 \]

Step 3: Combine the results

Now, we can combine the two results: \[ -7 < y \leq 5 \]

Step 4: Write the answer in compound inequality notation

In compound inequality notation, the answer is: \[ -7 < y \leq 5 \]

As integers, the solution can be expressed as: \[ y \in (-7, 5] \]

Final Answer: \((-7, 5]\) (or in interval notation: all integers \(y\) such that \(y > -7\) and \(y \leq 5\))