$3(6-4 x)<36-9 x$
Select the correct answer below and, if necessary, fill in the answer box to complete your choice.
A. The solution set expressed in interval notation is $\square$ .
(Simplify your answer. Use integers or fractions for any numbers in the expression.)
B. The solution set is $\varnothing$.
Answer (2)
Expand the left side of the inequality: 3 ( 6 − 4 x ) < 36 − 9 x becomes 18 − 12 x < 36 − 9 x .
Isolate x by adding 12 x to both sides: 18 < 36 + 3 x .
Subtract 36 from both sides: − 18 < 3 x .
Divide by 3 to find the solution: -6"> x > − 6 , which in interval notation is ( − 6 , ∞ ) .
Explanation
Understanding the Problem We are given the inequality 3 ( 6 − 4 x ) < 36 − 9 x . Our goal is to find the solution set for x , expressing it in interval notation if possible, or determining if the solution set is empty.
Expanding the Left Side First, we expand the left side of the inequality: 3 ( 6 − 4 x ) = 18 − 12 x .
Rewriting the Inequality Now we rewrite the inequality with the expanded form: 18 − 12 x < 36 − 9 x .
Adding 12x to Both Sides Next, we want to isolate x . We add 12 x to both sides of the inequality: 18 − 12 x + 12 x < 36 − 9 x + 12 x , which simplifies to 18 < 36 + 3 x .
Subtracting 36 from Both Sides Now, we subtract 36 from both sides: 18 − 36 < 36 + 3 x − 36 , which simplifies to − 18 < 3 x .
Dividing by 3 Finally, we divide both sides by 3: 3 − 18 < 3 3 x , which simplifies to − 6 < x . This is equivalent to -6"> x > − 6 .
Expressing the Solution Set in Interval Notation The solution set is all x such that -6"> x > − 6 . In interval notation, this is written as ( − 6 , ∞ ) .
Final Answer Therefore, the correct answer is A, and the solution set in interval notation is ( − 6 , ∞ ) .
Examples
Imagine you're planning a party and have a budget constraint. This inequality helps determine the maximum number of guests you can invite while staying within your budget. For example, if each guest costs a certain amount and you have fixed expenses, solving the inequality will tell you the upper limit on the number of guests. Inequalities are useful in various real-life scenarios like budgeting, resource allocation, and setting limits.
The solution to the inequality 3 ( 6 − 4 x ) < 36 − 9 x is -6"> x > − 6 . In interval notation, this is represented as ( − 6 , ∞ ) . Therefore, the correct choice is A, with the solution set being ( − 6 , ∞ ) .
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