The following inequality is given: \( 3(x+7)<\frac{x}{2}+1 \) Solve for \( x \) in the inequality and Represent your answer on a number line

To solve the inequality \( 3(x + 7) < \frac{x}{2} + 1 \), start by distributing on the left side: \( 3x + 21 < \frac{x}{2} + 1 \) Next, eliminate the fraction by multiplying the entire inequality by 2 (remember, this doesn’t change the direction of the inequality): \( 6x + 42 < x + 2 \) Now, move \( x \) to the left side and simplify: \( 6x - x + 42 < 2 \) This leads to: \( 5x + 42 < 2 \) Subtract 42 from both sides: \( 5x < 2 - 42 \) \( 5x < -40 \) Finally, divide both sides by 5: \( x < -8 \) Representing this on a number line, you would draw a line extending to the left from -8, with an open circle at -8 to indicate that it’s not included. --- Back in the day, inequalities were often a way to represent trade-offs, especially in economic discussions! Merchants used to compute their profits against costs and constraints, leading to the foundational concepts we utilize in modern mathematics. These inequalities guide business decisions, making them more than just numbers. In the world of real estate, knowing how to manipulate inequalities can help you understand property values! If a house is worth more than \( 200,000 \) but not exceeding \( 300,000 \), you can express this as \( 200,000 < x < 300,000 \). Understanding these ranges can make a significant difference in investment decisions, reinforcing the importance of inequalities in our everyday transactions!