Question

use the following situation to answer the following three - part question: cameron and joey are both saving money to buy a new video game. cameron already has $25, and he is saving $10 every week. joey has $40, and he is saving $5 every week.
(a) which inequality would you use to solve for the number of weeks ( w ) it would take cameron to have more saved money than joey?
options:
a) ( 25w + 10 < 40w + 5 )
b) ( 25 + 10w < 40 + 5w )
c) ( 25 + 10w > 40 + 5w )
d) ( 25w + 10 > 40w + 5 )
(b) solve the inequality (solution box)
(c) explain what your solution means in context to the situation (text box: my solution means that...)

Explanation:

Part (a)

To determine the inequality, we analyze Cameron's and Joey's savings:

• Cameron starts with $25 and saves $10 per week. So his total savings after \( w \) weeks is \( 25 + 10w \).
• Joey starts with $40 and saves $5 per week. So his total savings after \( w \) weeks is \( 40 + 5w \).

We want to find when Cameron has more saved than Joey, so we set up the inequality \( 25 + 10w > 40 + 5w \). This corresponds to option D.

Part (b)

Step 1: Subtract \( 5w \) from both sides

To isolate the variable terms, we subtract \( 5w \) from each side of the inequality \( 25 + 10w > 40 + 5w \).
\[
25 + 10w - 5w > 40 + 5w - 5w
\]
\[
25 + 5w > 40
\]

Step 2: Subtract 25 from both sides

Next, we subtract 25 from both sides to isolate the term with \( w \).
\[
25 + 5w - 25 > 40 - 25
\]
\[
5w > 15
\]

Step 3: Divide by 5

Finally, we divide both sides by 5 to solve for \( w \).
\[
\frac{5w}{5} > \frac{15}{5}
\]
\[
w > 3
\]

Part (c)

The solution \( w > 3 \) means that after more than 3 weeks (i.e., starting from the 4th week and beyond), Cameron will have more money saved than Joey.

Final Answers

(a) \(\boldsymbol{D}\)
(b) \(\boldsymbol{w > 3}\)
(c) After more than 3 weeks, Cameron has more saved than Joey.

Answer:

Part (a)

To determine the inequality, we analyze Cameron's and Joey's savings:

• Cameron starts with $25 and saves $10 per week. So his total savings after \( w \) weeks is \( 25 + 10w \).
• Joey starts with $40 and saves $5 per week. So his total savings after \( w \) weeks is \( 40 + 5w \).

We want to find when Cameron has more saved than Joey, so we set up the inequality \( 25 + 10w > 40 + 5w \). This corresponds to option D.

Part (b)

Step 1: Subtract \( 5w \) from both sides

To isolate the variable terms, we subtract \( 5w \) from each side of the inequality \( 25 + 10w > 40 + 5w \).
\[
25 + 10w - 5w > 40 + 5w - 5w
\]
\[
25 + 5w > 40
\]

Step 2: Subtract 25 from both sides

Next, we subtract 25 from both sides to isolate the term with \( w \).
\[
25 + 5w - 25 > 40 - 25
\]
\[
5w > 15
\]

Step 3: Divide by 5

Finally, we divide both sides by 5 to solve for \( w \).
\[
\frac{5w}{5} > \frac{15}{5}
\]
\[
w > 3
\]

Part (c)

The solution \( w > 3 \) means that after more than 3 weeks (i.e., starting from the 4th week and beyond), Cameron will have more money saved than Joey.

Final Answers

(a) \(\boldsymbol{D}\)
(b) \(\boldsymbol{w > 3}\)
(c) After more than 3 weeks, Cameron has more saved than Joey.