Answer :
Sure! Let's solve this problem step-by-step to find the appropriate inequality for determining the number of years, [tex]\( n \)[/tex], over which the tuition at Jen's school remains less than [tex]$20,000, given it increases by 5% per year.
1. Understand the Initial Condition:
- The current tuition is $[/tex]12,000.
- The tuition increases by 5% per year.
2. Determine the Formula for Tuition Over Time:
- If tuition increases by 5% each year, after one year, the tuition will be:
[tex]\[
\text{Next year's tuition} = \text{Current tuition} \times (1 + 0.05)
\][/tex]
This simplifies to:
[tex]\[
\text{Next year's tuition} = \text{Current tuition} \times 1.05
\][/tex]
- This pattern continues for each subsequent year.
3. Generalize the Formula for [tex]\( n \)[/tex] Years:
- The tuition after [tex]\( n \)[/tex] years is given by:
[tex]\[
\text{Tuition after } n \text{ years} = 12000 \times (1.05)^n
\][/tex]
4. Set Up the Inequality:
- We are asked to find when this tuition will be less than [tex]$20,000. Therefore, we set up the inequality:
\[
12000 \times (1.05)^n < 20000
\]
- This inequality reflects the condition we described: tuition that increases by 5% per year and must remain less than $[/tex]20,000.
5. Confirm the Correct Inequality:
- Among the given options, the inequality that matches our setup is:
[tex]\[
\boxed{C. \ 12000(1.05)^n < 20000}
\][/tex]
This is how you determine which inequality to use to solve for the number of years [tex]\( n \)[/tex] it takes for the tuition to remain below $20,000.
1. Understand the Initial Condition:
- The current tuition is $[/tex]12,000.
- The tuition increases by 5% per year.
2. Determine the Formula for Tuition Over Time:
- If tuition increases by 5% each year, after one year, the tuition will be:
[tex]\[
\text{Next year's tuition} = \text{Current tuition} \times (1 + 0.05)
\][/tex]
This simplifies to:
[tex]\[
\text{Next year's tuition} = \text{Current tuition} \times 1.05
\][/tex]
- This pattern continues for each subsequent year.
3. Generalize the Formula for [tex]\( n \)[/tex] Years:
- The tuition after [tex]\( n \)[/tex] years is given by:
[tex]\[
\text{Tuition after } n \text{ years} = 12000 \times (1.05)^n
\][/tex]
4. Set Up the Inequality:
- We are asked to find when this tuition will be less than [tex]$20,000. Therefore, we set up the inequality:
\[
12000 \times (1.05)^n < 20000
\]
- This inequality reflects the condition we described: tuition that increases by 5% per year and must remain less than $[/tex]20,000.
5. Confirm the Correct Inequality:
- Among the given options, the inequality that matches our setup is:
[tex]\[
\boxed{C. \ 12000(1.05)^n < 20000}
\][/tex]
This is how you determine which inequality to use to solve for the number of years [tex]\( n \)[/tex] it takes for the tuition to remain below $20,000.
