Which is the solution set of the compound inequality [tex]$3.5 x-10>-3$[/tex] and [tex]$8 x-9<39$[/tex]?
A. [tex]$-2<x<6$[/tex]
B. [tex]$-2 \leq x < 6$[/tex]
C. [tex]$2<x<6$[/tex]
D. [tex]$2 \leq x < 6$[/tex]
Answer (2)
Solve the first inequality -3"> 3.5 x − 10 > − 3 to get 2"> x > 2 .
Solve the second inequality 8 x − 9 < 39 to get x < 6 .
Find the intersection of the two solution sets, which is 2 < x < 6 .
The solution set is 2 < x < 6 .
Explanation
Understanding the Problem We are given the compound inequality -3"> 3.5 x − 10 > − 3 and 8 x − 9 < 39 . We need to find the solution set for x that satisfies both inequalities.
Solving the First Inequality First, let's solve the inequality -3"> 3.5 x − 10 > − 3 . Add 10 to both sides: 7"> 3.5 x > 7
Divide both sides by 3.5: \frac{7}{3.5} = 2"> x > 3.5 7 = 2
Solving the Second Inequality Now, let's solve the inequality 8 x − 9 < 39 . Add 9 to both sides: 8 x < 48
Divide both sides by 8: x < 8 48 = 6
Finding the Intersection We need to find the intersection of the solution sets 2"> x > 2 and x < 6 . This means x must be greater than 2 and less than 6. Therefore, the solution set is 2 < x < 6 .
Final Answer The solution set of the compound inequality is 2 < x < 6 .
Examples
Imagine you're planning a road trip. You need to drive more than 2 hours but less than 6 hours each day to avoid fatigue and reach your destination on time. This compound inequality helps you define the acceptable range of driving hours per day, ensuring a safe and efficient journey. Understanding compound inequalities can help you manage constraints and optimize your plans in various real-life situations.
The solution set for the compound inequality is 2 < x < 6 , which means x must be greater than 2 and less than 6. The correct answer choice is C. 2 < x < 6 . This is the range of x that satisfies both inequalities.
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