Solve the equation and inequalities. Write the solution sets in interval notation if possible. (a) \( (5 x-6)(x-5)=0 \) (b) \( (5 x-6)(x-5)<0 \) (c) \( (5 x-6)(x-5) \leq 0 \) (d) \( (5 x-6)(x-5)>0 \) (e) \( (5 x-6)(x-5) \geq 0 \)

To solve the equation and inequalities you've provided, let's first find the roots of the equation \( (5x - 6)(x - 5) = 0 \). **(a)** Set each factor to zero: 1. \( 5x - 6 = 0 \) → \( x = \frac{6}{5} \) 2. \( x - 5 = 0 \) → \( x = 5 \) The solutions to the equation are \( x = \frac{6}{5} \) and \( x = 5 \). In interval notation, this can be expressed as \( \{ \frac{6}{5}, 5 \} \). **(b)** Now, let's solve the inequality \( (5x - 6)(x - 5) < 0 \). We analyze the intervals determined by the roots: - Test intervals: \( (-\infty, \frac{6}{5})\), \( (\frac{6}{5}, 5) \), and \( (5, \infty) \). 1. Choose a number from \( (-\infty, \frac{6}{5})\), say \( x = 0 \): \( (5(0) - 6)(0 - 5) = (-6)(-5) > 0 \) → doesn't satisfy. 2. Choose from \( (\frac{6}{5}, 5) \), let's try \( x = 4 \): \( (5(4) - 6)(4 - 5) = (20 - 6)(-1) = 14(-1) < 0 \) → satisfies. 3. Choose from \( (5, \infty) \), let \( x = 6 \): \( (5(6) - 6)(6 - 5) = (30 - 6)(1) = 24 > 0 \) → doesn't satisfy. Thus, the solution for \( (5x - 6)(x - 5) < 0 \) is: \( \left( \frac{6}{5}, 5 \right) \). **(c)** For \( (5x - 6)(x - 5) \leq 0 \), we include the endpoints: The solutions are \( \left[ \frac{6}{5}, 5 \right] \). **(d)** For \( (5x - 6)(x - 5) > 0 \), we also consider the intervals: Using previous tests, it satisfies in: - The intervals \( (-\infty, \frac{6}{5}) \) and \( (5, \infty) \). Therefore, the solution here is: \( (-\infty, \frac{6}{5}) \cup (5, \infty) \). **(e)** Finally, for \( (5x - 6)(x - 5) \geq 0 \): The intervals where it is greater or equal to zero, including the roots, gives us: - The solution is: \( (-\infty, \frac{6}{5}] \cup [5, \infty) \). To summarize: - (a) \( \{ \frac{6}{5}, 5 \} \) - (b) \( \left( \frac{6}{5}, 5 \right) \) - (c) \( \left[ \frac{6}{5}, 5 \right] \) - (d) \( (-\infty, \frac{6}{5}) \cup (5, \infty) \) - (e) \( (-\infty, \frac{6}{5}] \cup [5, \infty) \)