Answer:
[tex]10 \leq t \leq 20[/tex]
Step-by-step explanation:
Given absolute value inequality:
[tex]1000(-2|t-15|+30) \geq 20000[/tex]
Isolate the absolute value on one side of the equation:
[tex]\implies \dfrac{1000-2|t-15|+30)}{1000} \geq \dfrac{20000}{1000}[/tex]
[tex]\implies -2|t-15|+30 \geq 20[/tex]
[tex]\implies -2|t-15|+30-30 \geq 20-30[/tex]
[tex]\implies -2|t-15|\geq -10[/tex]
[tex]\implies \dfrac{-2|t-15|}{-2}\geq \dfrac{-10}{-2}[/tex]
[tex]\implies |t-15|\leq 5[/tex]
[tex]\textsf{Apply absolute rule:\quad {If} $|u| \leq a, \;a > 0$ \;then\; $-a \leq u \leq a$}[/tex]
[tex]\implies -5 \leq t-15\leq 5[/tex]
Solve both equations:
[tex]\begin{aligned}\underline{\sf Equation\;1} && \underline{\sf Equation\;2}\\t-15 &\geq -5 \quad & t-15& \leq 5\\t-15+15& \geq-5+15 &\qquad t-15+15& \leq 5+15\\t&\geq10 & t &\leq20\end{aligned}[/tex]
Merge the overlapping intervals:
[tex]10 \leq t \leq 20[/tex]
