Question

Solve the equation: -2|x - 14| + 5 = -6|x - 14| - 1

Choose one of the following options:

A) x = -21 or x = 21
B) x = 25/2 or x = 31/2
C) x = -25/2 or x = -31/2
D) Only x = -3/2
E) There are no solutions

Answer 1

To solve the given equation, we simplify both sides and consider two cases: when the expression inside the absolute value is positive and when it is negative. By solving for m in each case and substituting back into the original equation, we find the solutions to be x = 0 and x = 16.

Step-by-step explanation:

To solve the equation -2|x - 14| + 5 = -6|x - 14| - 1, we first simplify both sides of the equation:

|-2m + 28| + 5 = |-6m + 84| - 1 ... (where m = x - 14)

Now, we can consider two cases:

1. Case 1: -2m + 28 is positive
2. Case 2: -2m + 28 is negative

Case 1: -2m + 28 is positive:

• -2m + 28 = -6m + 84
• 4m = -56
• m = -14

Case 2: -2m + 28 is negative:

• -(-2m + 28) + 5 = -(-6m + 84) - 1
• 2m - 28 + 5 = 6m - 84 - 1
• -4m = -8
• m = 2

Therefore, the solution to the equation is x = 14 + m:

• x = 14 + (-14) = 0
• x = 14 + 2 = 16

So, the correct answer is: x = 0 or x = 16.