To determine which scenarios result in a change in [tex]\(x\)[/tex] of 4, we will calculate [tex]\(\Delta x\)[/tex] for each given scenario. The change in [tex]\(x\)[/tex], [tex]\(\Delta x\)[/tex], is found by taking the absolute difference between the final value [tex]\(x_f\)[/tex] and the initial value [tex]\(x_i\)[/tex].
Let's go through each scenario step-by-step:
1. Scenario 1: [tex]\(\Delta x = x_f - x_i = -4 - (-8) = -4 + 8 = 4\)[/tex]
Here, [tex]\(\Delta x = 4\)[/tex], which matches the required change. So, this scenario is valid.
2. Scenario 2: [tex]\(\Delta x = x_f - x_i = 7 - 4 = 3\)[/tex]
Here, [tex]\(\Delta x = 3\)[/tex], which does not match the required change of 4. So, this scenario is not valid.
3. Scenario 3: [tex]\(\Delta x = x_f - x_i = 12 - 15 = -3\)[/tex]
Taking the absolute value, [tex]\(\Delta x = | -3 | = 3\)[/tex], which does not match the required change of 4. So, this scenario is not valid.
4. Scenario 4: [tex]\(\Delta x = x_f - x_i = 9 - 5 = 4\)[/tex]
Here, [tex]\(\Delta x = 4\)[/tex], which matches the required change. So, this scenario is valid.
5. Scenario 5: [tex]\(\Delta x = x_f - x_i = -8 - (-4) = -8 + 4 = -4\)[/tex]
Taking the absolute value, [tex]\(\Delta x = | -4 | = 4\)[/tex], which matches the required change. So, this scenario is valid.
Based on the calculations, the scenarios that produce the required change in [tex]\(x\)[/tex] of 4 are:
1. [tex]\(x\)[/tex] changes from [tex]\(x_i = -8\)[/tex] to [tex]\(x_f = -4\)[/tex]
2. [tex]\(x\)[/tex] changes from [tex]\(x_i = 5\)[/tex] to [tex]\(x_f = 9\)[/tex]
3. [tex]\(x\)[/tex] changes from [tex]\(x_i = -4\)[/tex] to [tex]\(x_f = -8\)[/tex]
Thus, the valid scenarios are 1, 4, and 5.
