In a linear number pattern, the first term is \( 2 x+11 \), the second term is 2 and the fourth term is \( 2 x-4 \). Calculate the value of \( x \).

In a linear number pattern, the difference between consecutive terms is constant. Let's denote the first term as \( a_1 = 2x + 11 \), the second term as \( a_2 = 2 \), and the fourth term as \( a_4 = 2x - 4 \). Since the difference between consecutive terms is constant, we can express this as: \[ a_2 - a_1 = a_3 - a_2 = a_4 - a_3 \] First, we calculate the difference \( a_2 - a_1 \): \[ a_2 - a_1 = 2 - (2x + 11) = 2 - 2x - 11 = -2x - 9 \] Next, we need to express \( a_3 \) in terms of \( a_2 \) and the common difference \( d \): \[ a_3 = a_2 + d = 2 + (-2x - 9) = -2x - 7 \] Now, we can find the difference \( a_4 - a_3 \): \[ a_4 - a_3 = (2x - 4) - (-2x - 7) = 2x - 4 + 2x + 7 = 4x + 3 \] Since the differences \( a_2 - a_1 \) and \( a_4 - a_3 \) must be equal, we set them equal to each other: \[ -2x - 9 = 4x + 3 \] Now, we solve for \( x \): 1. Add \( 2x \) to both sides: \[ -9 = 6x + 3 \] 2. Subtract \( 3 \) from both sides: \[ -12 = 6x \] 3. Divide by \( 6 \): \[ x = -2 \] Thus, the value of \( x \) is \[ \boxed{-2} \]