Solve $2 x+4 \leq 12$ or $9+12 x \geq 117$
Answer (2)
Solve the first inequality 2 x + 4 ≤ 12 , which gives x ≤ 4 .
Solve the second inequality 9 + 12 x ≥ 117 , which gives x ≥ 9 .
Combine the solutions using 'or', resulting in x ≤ 4 or x ≥ 9 .
The solution set is x ≤ 4 or x ≥ 9 , which in interval notation is ( − ∞ , 4 ] ∪ [ 9 , ∞ ) .
Explanation
Understanding the Problem We are given the compound inequality 2 x + 4 ≤ 12 or 9 + 12 x ≥ 117 . We need to solve each inequality separately and then combine the solutions using the 'or' condition. This means the solution set will include all values of x that satisfy either inequality.
Solving the First Inequality First, let's solve the inequality 2 x + 4 ≤ 12 . Subtract 4 from both sides: 2 x + 4 − 4 ≤ 12 − 4 2 x ≤ 8 Divide both sides by 2: 2 2 x ≤ 2 8 x ≤ 4
Solving the Second Inequality Next, let's solve the inequality 9 + 12 x ≥ 117 . Subtract 9 from both sides: 9 + 12 x − 9 ≥ 117 − 9 12 x ≥ 108 Divide both sides by 12: 12 12 x ≥ 12 108 x ≥ 9
Combining the Solutions Now, we combine the solutions. We have x ≤ 4 or x ≥ 9 . This means x can be any number less than or equal to 4, or any number greater than or equal to 9. In interval notation, this is ( − ∞ , 4 ] ∪ [ 9 , ∞ ) .
Final Answer Therefore, the solution to the compound inequality is x ≤ 4 or x ≥ 9 .
Examples
Imagine you are planning a road trip and need to decide which route to take based on the distance and time available. Suppose you have two options: Route A takes you through scenic towns but is only feasible if you drive no more than 4 hours a day ( x ≤ 4 ), or Route B, which is a highway route, is only worth it if you drive at least 9 hours a day ( x ≥ 9 ). This problem helps you decide which route to take based on your driving preferences and time constraints. Understanding compound inequalities helps in making decisions based on multiple conditions.
The solution to the inequalities 2 x + 4 ≤ 12 or 9 + 12 x ≥ 117 is x ≤ 4 or x ≥ 9 . In interval notation, this is expressed as ( − ∞ , 4 ] ∪ [ 9 , ∞ ) . These values represent all acceptable solutions for the given inequalities.
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