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

`10 \leq t \leq 20`

Explanation:

Given absolute value inequality:

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

Isolate the absolute value on one side of the equation:

`\implies (1000-2|t-15|+30))/(1000) \geq (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 (-2|t-15|)/(-2)\geq (-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`

Answer 2

• 10 ≤ t ≤ 20 or t ∈ [10; 20]

----------------------------

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]