Remember: |x| means either x for x >= 0 or it means -x for x < 0. We have to do that to your equation. Start with the t: t > |2t – 6| for t >=0 and -t > |2t – 6| for t < 0 Let's start with the first one: t > |2t – 6| for t >= 0

That means

t > 2t – 6 for t >= 0 and 2t-6 >=0

or

t > -(2t – 6) for t >= 0 and 2t – 6 < 0 Note that we now have two sets of three inequalities to solve for. Start with the first set: t > 2t – 6, t >= 0, 2t – 6 >= 0

6 > t , t >= 0; t >= 3

So we need numbers that are simultaneously less than 6, greater than or equal to 0 and greater than or equal to 3. For the second two, it’s necessary that the numbers be greater than or equal to 3. Add in the 6 > t and we have a solution set of

3 <= t < 6 Now the second one: t > -(2t – 6), t >= 0, 2t – 6 < 0 t > -2t + 6, t >= 0, t < 3 3t > 6, t >= 0, t < 3 t > 2, t >= 0, t < 3 This comes together making 2 < t < 3 Add that to our first solution and we have 2 < t < 6 Now we have to do the same thing with the second set. I'm going to let you do it, following this example. Please be aware that it's perfectly possible that the second set will yield no useful answers, so this may turn out to be our only solution, although I doubt it. I know it's an irritating process, but it really does have real-world applications.

or

l t l > l 6 l