To solve the inequalities \( h - 7 - 5h \leq -3 \) and \( -8h - 3 \geq h + 15 \), we will solve each inequality separately.
Solving the first inequality:
- Start with: \[ h - 7 - 5h \leq -3 \]
- Combine like terms: \[ -4h - 7 \leq -3 \]
- Add 7 to both sides: \[ -4h \leq 4 \]
- Divide by -4 (remember to reverse the inequality sign): \[ h \geq -1 \]
Solving the second inequality:
- Start with: \[ -8h - 3 \geq h + 15 \]
- Add \( 8h \) to both sides: \[ -3 \geq 9h + 15 \]
- Subtract 15 from both sides: \[ -18 \geq 9h \]
- Divide by 9: \[ -2 \geq h \quad \text{or} \quad h \leq -2 \]
Combining the results:
From the two inequalities, we have:
- \( h \geq -1 \)
- \( h \leq -2 \)
Since these two sets of results do not overlap, we write the solution in set notation as:
- \( h \geq -1 \) or \( h \leq -2 \)
Final answer:
The final solution, expressed as a compound inequality, is: \[ h \leq -2 \quad \text{or} \quad h \geq -1 \]