Question

1. if he uses 0.8 gb per day, how many days can he use the data before reaching the limit?
2. how much data will he have left after 12 days at the rate of 0.6 gb per day?

1v) a student works two jobs. job a pays $12 per hour; job b pays $15 per hour. she wants to earn at least $450 this week but can work no more than 35 hours total.

1. write a system of inequalities to represent the total hours and total earnings constraints.
2. if she works 20 hours at job a, how many hours must she work at job b to reach her earning goal?
3. what are all possible combinations of hours at job a and job b that meet both constraints?

v) a chemistry lab requires that a chemical solution be kept at a temperature between 15°c and 25°c for safe use. temperatures below 15°c or above 25°c can cause dangerous reactions.

1. represent the safe temperature range as an interval.
2. if the temperature is currently 12°c, how much must it increase to reach the safe range?
3. if the temperature is 28°c, is it within the safe range? if not, how much must it decrease?

Explanation:

IV

1.

Step1: Define variables

Let $x$ be the number of hours worked at Job A and $y$ be the number of hours worked at Job B.

Step2: Write total - hours constraint

The total number of hours worked is $x + y$, and it cannot exceed 35 hours. So, $x + y\leq35$.

Step3: Write total - earnings constraint

Job A pays $12 per hour and Job B pays $15 per hour, and the student wants to earn at least $450. So, $12x + 15y\geq450$.
The system of inequalities is

$$\begin{cases}x + y\leq35\\12x + 15y\geq450\end{cases}$$

, where $x\geq0$ and $y\geq0$ (since the number of hours cannot be negative).

Step1: First, find earnings from Job A

If she works 20 hours at Job A, her earnings from Job A are $12\times20 = 240$ dollars.

Step2: Then, find remaining earnings

She wants to earn at least $450$, so the remaining amount she needs to earn is $450 - 240=210$ dollars.

Step3: Calculate hours at Job B

Since Job B pays $15 per hour, the number of hours $y$ she needs to work at Job B is $\frac{210}{15}=14$ hours.

Step1: Rewrite inequalities

From $x + y\leq35$, we have $y\leq - x + 35$. From $12x+15y\geq450$, we can rewrite it as $y\geq-\frac{4}{5}x + 30$. Also, $x\geq0$ and $y\geq0$.

Step2: Find intersection points

Intersection of $y=-x + 35$ and $y =-\frac{4}{5}x+30$:
$-x + 35=-\frac{4}{5}x + 30$
$-x+\frac{4}{5}x=30 - 35$
$-\frac{1}{5}x=-5$, so $x = 25$ and $y=10$.
Intersection of $y=-x + 35$ and $x = 0$ gives $(0,35)$. Intersection of $y=-x + 35$ and $y = 0$ gives $(35,0)$. Intersection of $y=-\frac{4}{5}x + 30$ and $x = 0$ gives $(0,30)$. Intersection of $y=-\frac{4}{5}x + 30$ and $y = 0$ gives $(37.5,0)$ (but considering the non - negativity and the total - hours constraint, we ignore $x>35$).
The possible combinations are non - negative integer pairs $(x,y)$ such that $-\frac{4}{5}x + 30\leq y\leq - x + 35$, $x\geq0$ and $y\geq0$.
We can list the pairs by considering integer values of $x$ from 0 to 35:
For $x = 0$, $30\leq y\leq35$ (pairs $(0,30),(0,31),(0,32),(0,33),(0,34),(0,35)$);
For $x = 5$, $26\leq y\leq30$ (pairs $(5,26),(5,27),(5,28),(5,29),(5,30)$);
For $x = 10$, $22\leq y\leq25$ (pairs $(10,22),(10,23),(10,24),(10,25)$);
For $x = 15$, $18\leq y\leq20$ (pairs $(15,18),(15,19),(15,20)$);
For $x = 20$, $14\leq y\leq15$ (pairs $(20,14),(20,15)$);
For $x = 25$, $y = 10$ (pair $(25,10)$);
For $x$ from 26 to 35, $y$ values are non - existent or non - valid according to the inequalities.

Answer:

$$\begin{cases}x + y\leq35\\12x + 15y\geq450\\x\geq0\\y\geq0\end{cases}$$

2.