Solve for q.

Question

4+54q<83q+42≤54q+16
Write your answer as a compound inequality with integers, proper fractions, and improper fractions in simplest form.

Answer 1

To solve the compound inequality \( 4 + 54q < 83q + 42 \leq 54q + 16 \), we will break it into two parts and solve each part separately.

Part 1: Solve \( 4 + 54q < 83q + 42 \)

1. Subtract \(54q\) from both sides:

   \[ 4 < 83q - 54q + 42 \] \[ 4 < 29q + 42 \]

2. Subtract \(42\) from both sides:

   \[ 4 - 42 < 29q \] \[ -38 < 29q \]

3. Divide by \(29\) (note that the direction of the inequality does not change because \(29\) is positive):

   \[ -\frac{38}{29} < q \] We can write this as: \[ q > -\frac{38}{29} \]

Part 2: Solve \( 83q + 42 \leq 54q + 16 \)

1. Subtract \(54q\) from both sides:

   \[ 83q - 54q + 42 \leq 16 \] \[ 29q + 42 \leq 16 \]

2. Subtract \(42\) from both sides:

   \[ 29q \leq 16 - 42 \] \[ 29q \leq -26 \]

3. Divide by \(29\):

   \[ q \leq -\frac{26}{29} \]

Combine the Results

From Part 1, we have: \[ q > -\frac{38}{29} \]

From Part 2, we have: \[ q \leq -\frac{26}{29} \]

Thus, we combine the two results to get:

\[ -\frac{38}{29} < q \leq -\frac{26}{29} \]

Final Answer

\[ \boxed{-\frac{38}{29} < q \leq -\frac{26}{29}} \]