Solve $2(5 x+6)
A) $x<12$
B) There's no solution.
C) All real numbers
D) $x<-15$
Answer (1)
Expand the left side of the inequality: 2 ( 5 x + 6 ) = 10 x + 12 .
Combine like terms on the right side of the inequality: x − 15 + 9 x = 10 x − 15 .
Rewrite the inequality: 10 x + 12 < 10 x − 15 .
Subtract 10 x from both sides: 12 < − 15 , which is always false. Therefore, the answer is There’s no solution.
Explanation
Understanding the Inequality We are given the inequality 2 ( 5 x + 6 ) < x − 15 + 9 x . Our goal is to solve for x and determine which of the given options is correct.
Expanding the Left Side First, we expand the left side of the inequality: 2 ( 5 x + 6 ) = 10 x + 12
Combining Like Terms Next, we combine like terms on the right side of the inequality: x − 15 + 9 x = 10 x − 15
Rewriting the Inequality Now we rewrite the inequality with the simplified expressions: 10 x + 12 < 10 x − 15
Isolating the Constant Terms To solve for x , we subtract 10 x from both sides of the inequality: 10 x + 12 − 10 x < 10 x − 15 − 10 x 12 < − 15
Analyzing the Result The resulting inequality is 12 < − 15 . This statement is always false, regardless of the value of x . Therefore, there is no solution to the inequality.
Final Answer Comparing our result with the given options, we see that option B, "There's no solution," is the correct answer.
Examples
Imagine you're trying to determine if you can afford a certain number of items. The inequality represents a budget constraint where the left side is your expenses and the right side is your income. If simplifying the inequality leads to a false statement like 12 < − 15 , it means no matter how you adjust the number of items, your expenses will always be greater than your income, indicating that you cannot afford the items. This type of problem helps in making financial decisions and understanding constraints.
