Solve $5 x+12 \leq 32$ or $6+10 x \geq 96$
Answer (1)
Solve the first inequality: 5 x + 12 ≤ 32 which simplifies to x ≤ 4 .
Solve the second inequality: 6 + 10 x ≥ 96 which simplifies to x ≥ 9 .
Combine the solutions using 'or': x ≤ 4 or x ≥ 9 .
Express the solution in interval notation: ( − ∞ , 4 ] ∪ [ 9 , ∞ ) . The final answer is x ≤ 4 or x ≥ 9 .
Explanation
Analyze the problem We are given the compound inequality 5 x + 12 ≤ 32 or 6 + 10 x ≥ 96 . We need to solve each inequality separately and then combine the solutions.
Solve the first inequality First, let's solve the inequality 5 x + 12 ≤ 32 . Subtract 12 from both sides: 5 x + 12 − 12 ≤ 32 − 12
5 x ≤ 20
Divide both sides by 5: 5 5 x ≤ 5 20
x ≤ 4
Solve the second inequality Now, let's solve the inequality 6 + 10 x ≥ 96 . Subtract 6 from both sides: 6 + 10 x − 6 ≥ 96 − 6
10 x ≥ 90
Divide both sides by 10: 10 10 x ≥ 10 90
x ≥ 9
Combine the solutions The solution to the compound inequality is the union of the solutions to the individual inequalities. So, we have x ≤ 4 or x ≥ 9 . In interval notation, this is ( − ∞ , 4 ] ∪ [ 9 , ∞ ) .
Final Answer Therefore, the solution to the compound inequality 5 x + 12 ≤ 32 or 6 + 10 x ≥ 96 is x ≤ 4 or x ≥ 9 .
Examples
Imagine you're planning a party and need to decide how many snacks to buy. You know you'll have either a small group (up to 4 people) or a larger group (at least 9 people). This problem helps you determine the range of possible guest counts, ensuring you're prepared for either scenario. Understanding inequalities helps in real-life decision-making, such as budgeting, resource allocation, and planning for different possibilities.
