Which inequality represents all the solutions of $10(3 x+2)>7(2 x-4)?$
A. $x>-4$
B. $x<-4$
C. $x>-3$
D. $x<-3$
Answer (2)
Expand the inequality: 7(2x-4)"> 10 ( 3 x + 2 ) > 7 ( 2 x − 4 ) becomes 14x - 28"> 30 x + 20 > 14 x − 28 .
Combine like terms: Subtract 14 x from both sides to get -28"> 16 x + 20 > − 28 .
Isolate the x term: Subtract 20 from both sides to get -48"> 16 x > − 48 .
Solve for x: Divide both sides by 16 to find -3"> x > − 3 .
The solution is -3}"> x > − 3 .
Explanation
Understanding the Problem We are given the inequality 7(2x-4)"> 10 ( 3 x + 2 ) > 7 ( 2 x − 4 ) and we want to find the solution set for x . This means we want to isolate x on one side of the inequality to determine the values of x that satisfy the inequality.
Expanding the Inequality First, we expand both sides of the inequality using the distributive property: 7(2x-4)"> 10 ( 3 x + 2 ) > 7 ( 2 x − 4 ) 14x - 28"> 30 x + 20 > 14 x − 28
Combining Like Terms Next, we want to isolate the terms with x on one side of the inequality and the constant terms on the other side. We can subtract 14 x from both sides: 14x - 14x - 28"> 30 x − 14 x + 20 > 14 x − 14 x − 28 -28"> 16 x + 20 > − 28
Isolating the x Term Now, we subtract 20 from both sides to isolate the term with x :
-28 - 20"> 16 x + 20 − 20 > − 28 − 20 -48"> 16 x > − 48
Solving for x Finally, we divide both sides by 16 to solve for x :
\frac{-48}{16}"> 16 16 x > 16 − 48 -3"> x > − 3
Final Answer Therefore, the solution to the inequality is -3"> x > − 3 . This corresponds to option C.
Examples
Understanding inequalities is crucial in many real-world scenarios, such as budgeting and resource allocation. For instance, if you have a certain amount of money to spend on groceries and each item has a different cost, you can use inequalities to determine the maximum quantity of each item you can buy while staying within your budget. Similarly, in business, inequalities can help determine the minimum sales required to make a profit or the maximum expenses allowed to stay within a certain profit margin. These applications highlight the practical importance of mastering inequalities.
The solution to the inequality 7(2x-4)"> 10 ( 3 x + 2 ) > 7 ( 2 x − 4 ) is -3"> x > − 3 . Thus, the correct answer is option C.
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