Question

translate each phrase to an inequality. let x be the variable. (ex. 9 on notes)

1. a speed that is greater than 60 miles per hour.
2. an age that is at least 21 years old.
3. a salary that is more than $40,000.
4. a speed that does not exceed 70 miles per hour.
5. find all values of x so that the perimeter of the rectangle is less than 50 feet.
6. a student scores 74 out of 100 on a test. if the maximum score on the next test is also 100 points, what score does the student need to maintain at least an average of 80?
7. parking in a student lot costs $2 for the first half hour and $1.25 for each hour thereafter. a partial hour is charged the same as a full hour. what is the longest time that a student can park in this lot for $8?
8. if the temperature on the ground is 90°f, then the air temperature x miles high is given by t = 90 - 19x. determine the altitudes at which the air temperature is less than 45°f. (ex. 10 on notes)
9. the cost to produce one compact disc is $1.50 plus a one - time fixed cost of $2000. the revenue received from selling one compact disc is $12. (ex. 11 on notes)

a.) write a formula that gives the cost c of producing x compact discs.
b.) write a formula that gives the revenue r from selling x compact discs.
c.) profit equals revenue minus cost. write a formula that calculates the profit p from selling x compact discs

Explanation:

Step1: Translate 32

Let $x$ be the speed. "Greater than 60" gives $x>60$.

Step2: Translate 33

Let $x$ be the age. "At least 21" means $x\geq21$.

Step3: Translate 34

Let $x$ be the salary. "More than 40000" gives $x > 40000$.

Step4: Translate 35

Let $x$ be the speed. "Does not exceed 70" means $x\leq70$.

Step5: Translate 36

Let the length and width of the rectangle be $l$ and $w$. Perimeter $P = 2(l + w)$. If we assume some relationship with $x$, say $l=x$ and $w$ is a constant $a$, then $2(x + a)<50$, which simplifies to $x + a<25$ or $x<25 - a$.

Step6: Translate 37

Let the score on the next test be $x$. The average of the two - test scores is $\frac{74 + x}{2}$. To maintain at least an average of 80, we have $\frac{74+x}{2}\geq80$. Multiply both sides by 2: $74 + x\geq160$, then $x\geq160 - 74=86$.

Step7: Translate 38

Let the number of hours parked after the first half - hour be $x$. The cost function $C=2 + 1.25x$. We want to find $x$ when $C = 8$. So $2+1.25x=8$, $1.25x=6$, $x = 4.8$. The total time $t=0.5 + x$. The total time $t = 5.3$ hours. But since a partial hour is charged as a full hour, the longest time is 5 hours.

Step8: Translate 39

We have the temperature formula $T = 90-19x$. We want to find $x$ when $T<45$. So $90-19x<45$. Subtract 90 from both sides: $-19x<45 - 90=-45$. Divide both sides by - 19 and reverse the inequality sign: $x>\frac{45}{19}\approx2.37$.

Step9: Translate 40a

The cost $C$ of producing $x$ compact discs is $C = 1.5x+2000$.

Step10: Translate 40b

The revenue $R$ from selling $x$ compact discs is $R = 12x$.

Step11: Translate 40c

The profit $P$ is $P=R - C=12x-(1.5x + 2000)=12x-1.5x-2000 = 10.5x-2000$.

Answer:

1. $x>60$
2. $x\geq21$
3. $x > 40000$
4. $x\leq70$
5. Depends on rectangle's other side; if $P = 2(x + a)<50$, then $x<25 - a$
6. $x\geq86$
7. 5 hours
8. $x>\frac{45}{19}$

40a. $C = 1.5x+2000$
40b. $R = 12x$
40c. $P=10.5x - 2000$