Question 1083260
break these up into 2 separate inequalities.


start with 2/3 * t - 1 < t + 7 <= -2t + 15


the first set of inequalities is:


2/3 * t - 1 < t + 7
add 1 to both sides to get:
2/3 * t < t + 8
subtract t from both sides to get:
2/3 * t - t < 8
simplify to get:
-1/3 * t < 8
multiply both sides by 3 to get:
-t < 24
multiply both sides by -1 to get:
t > -24
multiplying both sides by -1 reverses the inequality.
solution for the first set of inequalities is t > -24


the second set of inequalities is:


t + 7 <= -2 * t + 15
subtract 7 from both sides to get:
t <= -2 * t + 8
add 2t to both sides to get:
3 * t <= 8
divide both sides by 3 to get:
t <= 8/3
solution for the second set of inequalities is t <= 8/3


your solution is that t > -24 and t <= 8/3


this can be written as -24 < t <= 8/3


in interval notation, this would be written as:


(-24,8/3]


you can test this out with various values of t to ensure the inequality is correct.


the inequality statement to test is:


2/3 * t - 1 < t + 7 <= -2t + 15


in order for the statement to be true, all parts of the inequality must be true.


2/3 * t - 1 must be smaller than t + 7 AND t + 7 must be smaller than or equal to -2t + 15


i checked and it all looks good, so i'm reasonably certain that the solution is good.