Which of the following values are solutions to the inequality $9 \leq 3 x+4 ?$ I. $-7$ II. $8$ III. $2$
Answer (2)
Substitute each value into the inequality 9 ≤ 3 x + 4 .
For x = − 7 , the inequality becomes 9 ≤ − 17 , which is false.
For x = 8 , the inequality becomes 9 ≤ 28 , which is true.
For x = 2 , the inequality becomes 9 ≤ 10 , which is true. Thus, the solutions are II and III, so the answer is II and III .
Explanation
Understanding the Problem We are given the inequality 9 ≤ 3 x + 4 , and we need to determine which of the values -7, 8, and 2 are solutions to this inequality. To do this, we will substitute each value for x and check if the inequality holds true.
Testing the Values Let's test each value:
I. For x = − 7 , we substitute into the inequality: 9 ≤ 3 ( − 7 ) + 4 . This simplifies to 9 ≤ − 21 + 4 , which is 9 ≤ − 17 . This statement is false, so -7 is not a solution.
II. For x = 8 , we substitute into the inequality: 9 ≤ 3 ( 8 ) + 4 . This simplifies to 9 ≤ 24 + 4 , which is 9 ≤ 28 . This statement is true, so 8 is a solution.
III. For x = 2 , we substitute into the inequality: 9 ≤ 3 ( 2 ) + 4 . This simplifies to 9 ≤ 6 + 4 , which is 9 ≤ 10 . This statement is true, so 2 is a solution.
Conclusion From our analysis, we found that 8 and 2 are solutions to the inequality, while -7 is not. Therefore, the correct answer is II and III.
Examples
Linear inequalities are used in everyday life to determine things like budget constraints. For example, if you have a budget of $50 for groceries and you want to buy apples that cost $2 per pound and bananas that cost 1 p er p o u n d , yo u c an se t u p anin e q u a l i t y t ore p rese n tt h e p oss ib l eco mbina t i o n so f a ppl es an d banana syo u c anb u y . I f x re p rese n t s t h e n u mb ero f p o u n d so f a ppl es an d y$ represents the number of pounds of bananas, the inequality would be 2 x + y ≤ 50 . This helps you determine how much of each fruit you can purchase without exceeding your budget.
The solutions to the inequality 9 ≤ 3 x + 4 are the values 8 and 2. The value -7 is not a solution. Therefore, the correct answer is II and III.
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