To solve this problem, let's analyze the information step by step.
1. Understanding the Ages:
- Jessica is 5 years older than her sister Jenna.
- Let's denote Jenna's current age as [tex]\( x \)[/tex].
2. Express Jessica's Current Age:
- Since Jessica is 5 years older than Jenna, we can write:
[tex]\[ \text{Jessica's age} = x + 5 \][/tex]
3. Ages in 5 Years:
- In 5 years, Jenna's age will be:
[tex]\[ x + 5 \][/tex]
- In 5 years, Jessica's age will be:
[tex]\[ (x + 5) + 5 = x + 10 \][/tex]
4. Jessica's Age 5 Years Ago:
- 5 years ago, Jessica's age was:
[tex]\[ (x + 5) - 5 = x \][/tex]
5. The Given Condition:
- Jenna tells Jessica that in 5 years, she (Jenna) will be as old as Jessica was 5 years ago.
- This age condition can be written as:
[tex]\[ \text{Jenna's age in 5 years} = \text{Jessica's age 5 years ago} \][/tex]
[tex]\[
x + 5 = x
\][/tex]
Now let's analyze the answer choices:
A. [tex]\( x + 5 = (5 - x) - 5 \)[/tex], which has one solution:
- This option does not match the condition we derived.
B. [tex]\( x + 5 = (x + 5) - 5 \)[/tex], which has infinitely many solutions:
- Simplify the right-hand side: [tex]\( (x + 5) - 5 = x \)[/tex]
- We get: [tex]\( x + 5 = x \)[/tex]
- This equation simplifies to an identity, which means it is always true regardless of the value of [tex]\( x \)[/tex]. Therefore, this choice is correct and it has infinitely many solutions.
C. [tex]\( x + 5 = (x + 5) - 5 \)[/tex], which has no solution:
- As we analyzed above, this equation [tex]\( x + 5 = x \)[/tex] has infinitely many solutions, not no solution. So this option is incorrect.
D. [tex]\( x + 5 = (5 - x) - 5 \)[/tex], which has infinitely many solutions:
- This option does not match the condition we derived and the equation is not equivalent to [tex]\( x + 5 = x \)[/tex].
E. [tex]\( (x + 5) + 5 = (x + 5) + 5 \)[/tex], which has infinitely many solutions:
- This equation is not relevant to the condition given in the problem.
After analyzing all of the options, the correct answer is:
B. [tex]\( x + 5 = (x + 5) - 5 \)[/tex], which has infinitely many solutions.
