What is the value of $x$ in the linear inequality?
$-3(4 x-8.2)<-11.98 x+14 \frac{3}{4}$
A. $x<49.25$
B. $x>49.25$
C. $x<492.5$
D. $x>492.5$
Answer (2)
Distribute the constant: − 12 x + 24.6 < − 11.98 x + 14.75
Combine like terms: − 0.02 x < − 9.85
Divide by the coefficient of x (and flip the inequality sign since we're dividing by a negative number): 492.5"> x > 492.5
The solution to the inequality is 492.5}"> x > 492.5
Explanation
Understanding the Inequality We are given the linear inequality − 3 ( 4 x − 8.2 ) < − 11.98 x + 14 4 3 . Our goal is to isolate x and determine the solution set.
Distributing the Constant First, distribute the − 3 on the left side of the inequality: − 3 ( 4 x ) − 3 ( − 8.2 ) < − 11.98 x + 14 4 3 , which simplifies to − 12 x + 24.6 < − 11.98 x + 14 4 3 .
Converting to Decimal Next, convert the mixed number 14 4 3 to a decimal: 14 4 3 = 14 + 4 3 = 14 + 0.75 = 14.75 . So the inequality becomes − 12 x + 24.6 < − 11.98 x + 14.75 .
Combining x Terms Now, add 11.98 x to both sides of the inequality to get the x terms on one side: − 12 x + 11.98 x + 24.6 < − 11.98 x + 11.98 x + 14.75 , which simplifies to − 0.02 x + 24.6 < 14.75 .
Isolating x Term Subtract 24.6 from both sides of the inequality to isolate the x term: − 0.02 x + 24.6 − 24.6 < 14.75 − 24.6 , which simplifies to − 0.02 x < − 9.85 .
Solving for x Finally, divide both sides of the inequality by − 0.02 . Remember that when dividing by a negative number, we must reverse the inequality sign: \frac{-9.85}{-0.02}"> − 0.02 − 0.02 x > − 0.02 − 9.85 , which simplifies to 492.5"> x > 492.5 .
Final Answer Therefore, the solution to the inequality is 492.5"> x > 492.5 . Comparing this to the given options, we see that the correct answer is D.
Examples
Linear inequalities are used in various real-life situations, such as budgeting and resource allocation. For example, if you have a certain amount of money to spend on groceries and each item has a different price, you can use a linear inequality to determine how many of each item you can buy without exceeding your budget. Similarly, in manufacturing, linear inequalities can help determine the optimal production levels to maximize profit while staying within resource constraints. Understanding and solving linear inequalities is a fundamental skill for making informed decisions in everyday life.
To solve the inequality − 3 ( 4 x − 8.2 ) < − 11.98 x + 14.75 , we find that 492.5"> x > 492.5 . Thus, the answer is D: 492.5"> x > 492.5 .
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