Final answer:
To solve the given inequality, we can expand and simplify the expression, set it equal to zero, find the critical points, and determine the interval where the expression is greater than or less than zero. The solution in interval notation is (-∞, 76) and (76, ∞), and in inequality notation, it is x < 76 and x > 76.
Step-by-step explanation:
The given inequality is 1/2(x-1)²-75(x-3)+4. To solve this inequality, we can start by expanding and simplifying the expression:
1/2(x-1)² - 75(x-3) + 4
= 1/2(x² - 2x + 1) - 75(x-3) + 4
= 1/2x² - x + 1/2 - 75x + 225 + 4
= 1/2x² - 76x + 230.5
Next, we can set this expression equal to zero and find the critical points:
1/2x² - 76x + 230.5 = 0
Now, we can solve this quadratic equation. However, since the question asks for the solution in interval notation, we can skip the quadratic formula and proceed to find the interval where the expression is greater than zero or less than zero.
To do this, we can find the vertex of the parabola defined by the quadratic expression. The x-coordinate of the vertex can be found using the formula x = -b/(2a). In this case, a = 1/2 and b = -76. Plugging in these values, we have x = -(-76)/(2 * 1/2) = 76. So the x-coordinate of the vertex is 76.
Since the coefficient of the x² term is positive, the parabola opens upwards and the vertex represents the minimum point. This means that the expression is greater than zero for x < 76 and is less than zero for x > 76.
Therefore, the solution to the inequality in interval notation is (-∞, 76) and (76, ∞). In inequality notation, it can be written as x < 76 and x > 76.