Solve First Inequality
Begin by solving the first inequality:
color(red)(3)(t - 3) + 1 >= 7
(color(red)(3) xx t) - (color(red)(3) xx 3) + 1 >= 7
3t - 9 + 1 >= 7
3t - 8 >= 7
3t - 8 + color(red)(8) >= 7 + color(red)(8)
3t - 0 >= 15
3t >= 15
(3t)/color(red)(3) >= 15/color(red)(3)
(color(red)(cancel(color(black)(3)))t)/cancel(color(red)(3)) >= 5
t >= 5
Solve Second Inequality
Next, we can solve the second inequality for t:
color(red)(2)(t + 1) + 3 <= 1
(color(red)(2) xx t) + (color(red)(2) xx 1) + 3 <= 1
2t + 2 + 3 <= 1
2t + 5 <= 1
2t + 5 - color(red)(5) <= 1 - color(red)(5)
2t + 0 <= -4
2t <= -4
(2t)/color(red)(2) <= -4/color(red)(2)
(color(red)(cancel(color(black)(2)))t)/cancel(color(red)(2)) <= -2
t <= -2
**The Solution Is:
t < -2; t >= 5
Or, in interval notation:
(-oo, -2]; [5, +oo)
