Question

Andrea's record label released her new album. Andrea wants to know when she has sales of at least $20,000 per week. She uses the related equation below to determine when sales will be at least this amount, where t represents time in weeks.

please finish the other half

Answer 1

Answer:

• 10 ≤ t ≤ 20 or t ∈ [10; 20]

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The other half is:

• t - 15 ≥ - 5
• t  ≥ - 5 + 15
• t  ≥ 10

The answer, considering both halves, is:

• 10 ≤ t ≤ 20

or

• t ∈ [10; 20]

Answer 2

Answer:

$10 \leq t \leq 20$

Step-by-step explanation:

Given absolute value inequality:

$1000(-2|t-15|+30) \geq 20000$

Isolate the absolute value on one side of the equation:

$\implies \dfrac{1000-2|t-15|+30)}{1000} \geq \dfrac{20000}{1000}$

$\implies -2|t-15|+30 \geq 20$

$\implies -2|t-15|+30-30 \geq 20-30$

$\implies -2|t-15|\geq -10$

$\implies \dfrac{-2|t-15|}{-2}\geq \dfrac{-10}{-2}$

$\implies |t-15|\leq 5$

$\textsf{Apply absolute rule:\quad {If} |u| \leq a, \;a > 0 \;then\; -a \leq u \leq a }$

$\implies -5 \leq t-15\leq 5$

Solve both equations:

$\begin{aligned}\underline{\sf Equation\;1} && \underline{\sf Equation\;2}\\t-15 &\geq -5 \quad & t-15& \leq 5\\t-15+15& \geq-5+15 &\qquad t-15+15& \leq 5+15\\t&\geq10 & t &\leq20\end{aligned}$

Merge the overlapping intervals:

$10 \leq t \leq 20$