f) \( x^{2}-7 x+6 \leq 0 \)

To solve the inequality \( x^{2}-7x+6 \leq 0 \), first factor the quadratic expression. It can be factored as \( (x - 1)(x - 6) \leq 0 \). Next, identify the critical points by setting each factor to zero, which gives us \( x = 1 \) and \( x = 6 \). Now, test intervals around these points: choose a number less than 1 (like 0), between 1 and 6 (like 3), and greater than 6 (like 7). You'll find that the expression is negative or zero between 1 and 6, including the endpoints. Therefore, the solution to the inequality is \( 1 \leq x \leq 6 \). In real-world scenarios, this quadratic inequality can represent constraints in optimization problems, such as determining feasible production levels where cost or output meets certain criteria. For instance, if x represents the quantity of products produced, understanding these constraints helps businesses maximize profit while minimizing costs effectively!