Final answer:
To find the x values that satisfy the inequality, we analyze intervals on the number line around the points x = -2 and x = 1, where the expressions inside the absolute value brackets are zero. We then solve the inequality within those intervals, taking into account the properties of absolute values.
Step-by-step explanation:
To find all x € R that satisfy the inequality 4 < |x + 2| + |x - 1| < 5, we need to consider the different cases of the absolute values depending on whether the expressions within the absolute value brackets are positive or negative. This is because the absolute value of a number is the non-negative value of that number without regard to its sign.
There are two key points to consider which will split the number line into three intervals. These points are where the expressions inside the absolute values equal zero: x = -2 and x = 1. We then examine the inequality in each of the intervals (-∞, -2), (-2, 1), and (1, ∞).
By solving the inequality in each of these intervals, we can find the range of x values that satisfy the equation. In this process, we may need to use algebraic manipulation and consider the definition of the absolute value function.