How to solve the values of t?

|t/2+4| + 3 > 2t

1 Answer

Consider the two separate positive and negative cases of the absolute value and then combine the two separate results

Explanation:

We can remove the absolute value sign by splitting the expression into more than one range according to when and when it doesn't change the sign of what's inside.

|t/2+4|+3>2t

First case - positive

If t/2+4>=0, then the inequality asks for t/2+4+3>2t, which rearranges to -3/2t> -7. Divide through by the t coefficient, noticing that the change of sign needs a matching switch of inequality direction:
t<14/3.

When t/2+4>=0, t>=-8. So for this first case we have specified a range of possible t values:
-8<=t<14/3.

Second case - negative

If t/2+4<0, then the inequality asks for -t/2-4+3>2t, which rearranges to -5/2t> 1. Divide through by the t coefficient, noticing that the change of sign needs a matching switch of inequality direction:
t<-2/5.

When t/2+4<0, t<-8. So for this second case we've specified two conditions that both point in the same direction. For them both to hold we take the strongest one, t<-8.

Combine cases

We now have two ranges of t that fit the inequality, t<-8 and -8<=t<14/3. In general it is perfectly acceptable to finish an answer to such a question by finding multiple ranges of validity. However, in this case, the two ranges can be combined because they touch each other at one end. So our final answer is:

t<14/3

Sanity check the answer by comparing graphs of both sides of the inequality:
graph{(y-(|x/2+4|+3))(y-2x)=0 [-40, 40, -20, 20]}