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\(\frac{2}{3} t - 1 < t + 7 \le -2t + 15 \)

We first have  the inequality

2/3 t - 1 < t + 7    subtract 7 from both sides

2/3 t  -  8 <  t     subtract 2/3 t  from both sides

-8 < 1/3 t     multiply both sides by 3

-24 <  t

And we have the second inequalty

t + 7 ≤  -2t + 15       add 2t  to both sides...subtract 7 from both sides

3t ≤  8      divide both sides by 3

t ≤ 8/3

So      (-24, 8/3 ]    is the solution