It is given that |x+3a| = 5a, where a is a positive constant. Find, in terms of a, the possible values of |x+7a| - |x-7a|.
Firstly, we need to find x in terms of a, which we can do this by using the given fact that |x+3a| = 5a. There are 2 methods of doing this; the first is by squaring both sides and solving the resulting quadratic, while the second is by equating the different possible values of the left hand side of the equation to the right hand side. The 'squaring both sides' method can be done in the following manner: (x+3a)2 = (5a)2. x2+3ax+3ax+9a2 = 25a2. x2+6ax+9a2 = 25a2. x2+6ax-16a2 = 0. (x-2a)(x+8a) = 0. Therefore, x = 2a or x = -8a. This method works because squaring either value that a modulus can take gives the same answer. For example |2| = 2 or -2. 2 and -2 squared both give 4. So we can square what's inside the modulus, 2, to get the same answer. Alternatively, the 'equating' method can be done in the following manner: x+3a = 5a or -(x+3a) = 5a. x = 2a or -x-3a = 5a. Therefore x = 2a or x = -8a (as shown in the above method). This method works because we know that the positive and negative values of the term inside the modulus on the left hand side of the equation both equate to the constant on the right hand side of the equation. This means we can form two separate equations and solve them. The other half of this question consists of finding the possible values of |x+7a| - |x-7a|. Now that we have values for x in terms of a, we can substitute them into the expression to find the possible values in terms of a. We know that a modulus sign means to take the actual size of the value, irrespective of the sign, essentially meaning that the positive value of the expression should be taken. Using this knowledge, we can arrive at our answer. Using x = 2a, we get: |2a+7a| - |2a-7a|. |9a| - |-5a|. 9a - 5a = 4a. Using x = -8a, we get: |-8a+7a| - |-8a-7a|. |-a| - |-15a|. a - 15a = -14a. So the answers to this question are 4a and -14a.
