QUESTION 1 \( 1.1 \quad \) Solve for \( x \) : \( 1.1 .1 \quad(2 x-3)(x+7)=0 \)

To solve the equation \( (2x - 3)(x + 7) = 0 \), you can set each factor equal to zero. This gives us two equations to solve: \( 2x - 3 = 0 \) and \( x + 7 = 0 \). From \( 2x - 3 = 0 \), adding 3 to both sides, we get \( 2x = 3 \), which gives \( x = \frac{3}{2} \). From \( x + 7 = 0 \), subtracting 7 from both sides gives us \( x = -7 \). Thus, the solutions are \( x = \frac{3}{2} \) and \( x = -7 \). Now let's explore how this equation relates to real-world applications. In practical scenarios, equations like this can be used in economic modeling. For instance, if you’re looking at profit and loss in a business, the factors may represent different components of your revenue model, such as price per item or quantities sold. Setting these components to zero allows businesses to analyze break-even points and make informed decisions. If you’re ever faced with multiplying factors like this, remember a classic pitfall: don’t forget to check both possibilities when you break it down to zero. Many students often stop at one factor and miss the potential solutions from the other! Always simplify and consider all roots for a full understanding of the problem.