a How much money will the studio cost and how much do they have to spend?
B
b Use inverse operations on all parts on the compound inequality.
C
c Substitute 6 in for the value of n.
A
a See solution.
B
bInequality: 0≤ n≤ 4
Considerations? You cannot record a negative amount of time.
C
c $155
Practice makes perfect
a Let's look at the different information given in the exercise and apply it to the inequality.
They will pay $225 to use the studio for one day.
They will pay $80 per hour for sound technicians.
We can think of the daily fee as a flat rate that we have to add one time to the hourly rate costs. Therefore, given that the variable n represents the number of hours the band uses the studio, we can write an expression for the amount of money they can spend.
225+80n
Since this is the amount of money they can spend, we need to look at their monetary restrictions. Since Jenna already has $200, this is the least amount that they can spend. However, they expect to raise $350 in pre-orders so they can spend up to $550 for the studio.
200+350=$550
The word least indicates greater than or equal to, and up to indicates less than or equal to. When all the parts are combined, you get the inequality given in the exercise.
200≤ 225+80n≤550
b In order to solve the inequality, we will need to use inverse operations on all parts of the inequality. Let's begin by separating the compound inequality into two simple inequalities.
200≤225+80n≤550
⇕
200≤225+80n & (I) 225+80n≤550 & (II)
We can solve these cases one at a time and then recombine them. We will start by reducing Inequality (I).
We can recombine these two inequalities to get a compound inequality.
-0.31≤ n ≤4.06
Be aware that you cannot record for a negative amount of time. The least amount of time spent in the studio will be 0. Also, people usually purchase hourly rentals by the hour, not by the minute. Since they cannot afford to purchase 5 hours, they can record up to 4 hours. Let's rewrite the inequality to fit the context of the problem.
0≤ n≤ 4
c We know that Jenna's band can record up to 4 hours. We need to see how much more money is needed if they want to purchase 6 hours of recording time. Remember that we already have an expression to find the total cost.
225+80nLet's substitute 6 into the expression and simplify.
Since they already have $550, we need to take away $550 from $705 to find the difference.
705-550=$155
Thus, they need $155 more to record for 6 hours.
