Question
Sarah is hiking. She starts at 80 feet and hikes up at a constant rate of 4 feet per minute. How many minutes could she be hiking if she is above 270 feet in elevation?
Write an inequality to represent the situation. Use x to represent the number of minutes.
80+4x>270
Good work!
You just wrote an inequality to represent the situation. Now solve the associated equation, 80 + 4x = 270, to find the boundary for the inequality. Show your work or solve on paper.
80 + 4x > 270
80+4x
=
270
x
=
47.5
Nice!
Now figure out if the boundary, x = 47.5, is a solution to the inequality. Does x = 47.5 make the inequality true or false?
80 + 4x > 270
True
False
Good work!
You found that x= 47.5 makes the inequality false. Now test x= 47 and x= 48. Do they make the inequality true or false?
x= 47x=47.5 (boundary)x= 4880 + 4x > 270
False
True
False
False
True
False
True
Nice!
Use the table to help you write the solution to the inequality.
x= 47x=47.5 (boundary)x= 4880 + 4x > 270FalseFalseTrue
x
≥
<
>
≤
Write an inequality to represent the situation. Use x to represent the number of minutes.
80+4x>270
Good work!
You just wrote an inequality to represent the situation. Now solve the associated equation, 80 + 4x = 270, to find the boundary for the inequality. Show your work or solve on paper.
80 + 4x > 270
80+4x
=
270
x
=
47.5
Nice!
Now figure out if the boundary, x = 47.5, is a solution to the inequality. Does x = 47.5 make the inequality true or false?
80 + 4x > 270
True
False
Good work!
You found that x= 47.5 makes the inequality false. Now test x= 47 and x= 48. Do they make the inequality true or false?
x= 47x=47.5 (boundary)x= 4880 + 4x > 270
False
True
False
False
True
False
True
Nice!
Use the table to help you write the solution to the inequality.
x= 47x=47.5 (boundary)x= 4880 + 4x > 270FalseFalseTrue
x
≥
<
>
≤
Answers
GPT-4o mini
To summarize the findings based on the inequality we derived and the corresponding calculations:
**Inequality:**
\[ 80 + 4x > 270 \]
**Solving the associated equation:**
\[ 80 + 4x = 270 \]
Subtract 80 from both sides:
\[ 4x = 190 \]
Divide by 4:
\[ x = 47.5 \]
**Boundary Check:**
When testing the boundary \( x = 47.5 \):
\[ 80 + 4(47.5) = 80 + 190 = 270 \]
This shows that at \( x = 47.5 \), the inequality \( 80 + 4x > 270 \) is false.
**Testing values:**
- For \( x = 47 \):
\[ 80 + 4(47) = 80 + 188 = 268 \] (which is less than 270) → **False**
- For \( x = 48 \):
\[ 80 + 4(48) = 80 + 192 = 272 \] (which is greater than 270) → **True**
**Results Summary:**
- \( x = 47 \) → False
- \( x = 47.5 \) → False (boundary)
- \( x = 48 \) → True
**Conclusion:**
Since we need \( 80 + 4x > 270 \) to be true, the solution to the inequality indicates the values of \( x \) that satisfy it:
**The solution to the inequality:**
\[ x > 47.5 \]
In interval notation, this can be represented as:
\[ x \in (47.5, \infty) \]
Thus, Sarah could be hiking for any time greater than 47.5 minutes to be above 270 feet in elevation.
**Inequality:**
\[ 80 + 4x > 270 \]
**Solving the associated equation:**
\[ 80 + 4x = 270 \]
Subtract 80 from both sides:
\[ 4x = 190 \]
Divide by 4:
\[ x = 47.5 \]
**Boundary Check:**
When testing the boundary \( x = 47.5 \):
\[ 80 + 4(47.5) = 80 + 190 = 270 \]
This shows that at \( x = 47.5 \), the inequality \( 80 + 4x > 270 \) is false.
**Testing values:**
- For \( x = 47 \):
\[ 80 + 4(47) = 80 + 188 = 268 \] (which is less than 270) → **False**
- For \( x = 48 \):
\[ 80 + 4(48) = 80 + 192 = 272 \] (which is greater than 270) → **True**
**Results Summary:**
- \( x = 47 \) → False
- \( x = 47.5 \) → False (boundary)
- \( x = 48 \) → True
**Conclusion:**
Since we need \( 80 + 4x > 270 \) to be true, the solution to the inequality indicates the values of \( x \) that satisfy it:
**The solution to the inequality:**
\[ x > 47.5 \]
In interval notation, this can be represented as:
\[ x \in (47.5, \infty) \]
Thus, Sarah could be hiking for any time greater than 47.5 minutes to be above 270 feet in elevation.