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
True
False
False
False
True
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
True
False
False
False
True
Answers
GPT-4o mini
Let's clarify the inequality you've set up and then check the values you've calculated.
You correctly set up the inequality to find the number of minutes she can hike for her elevation to be above 270 feet:
\[ 80 + 4x > 270 \]
Next, solving the associated equation:
\[ 80 + 4x = 270 \]
Subtract 80 from both sides:
\[ 4x = 270 - 80 \]
\[ 4x = 190 \]
Now, divide by 4:
\[ x = \frac{190}{4} \]
\[ x = 47.5 \]
Now, let's check this value in the original inequality. Since \(47.5\) is the boundary point that results in the elevation of exactly 270 feet:
1. Check for \( x = 47.5 \):
\[
80 + 4(47.5) = 80 + 190 = 270
\]
Here, \( 80 + 4(47.5) > 270 \) is **false** since it equals 270.
2. Now check \( x = 47 \):
\[
80 + 4(47) = 80 + 188 = 268
\]
Thus, \( 80 + 4(47) > 270 \) is **false**.
3. Check \( x = 48 \):
\[
80 + 4(48) = 80 + 192 = 272
\]
Here, \( 80 + 4(48) > 270 \) is **true**, since 272 is greater than 270.
In summary:
- \( x = 47.5 \) is the boundary, yielding exactly 270 feet, which **does not satisfy** the inequality.
- \( x = 47 \) results in 268 feet, which is **not sufficient** to satisfy the inequality.
- \( x = 48 \) produces 272 feet, which **does satisfy** the inequality.
So, the solutions to the inequality \( 80 + 4x > 270 \) are \( x > 47.5 \). Therefore, Sarah can hike for any number of minutes greater than 47.5 to be above 270 feet.
You correctly set up the inequality to find the number of minutes she can hike for her elevation to be above 270 feet:
\[ 80 + 4x > 270 \]
Next, solving the associated equation:
\[ 80 + 4x = 270 \]
Subtract 80 from both sides:
\[ 4x = 270 - 80 \]
\[ 4x = 190 \]
Now, divide by 4:
\[ x = \frac{190}{4} \]
\[ x = 47.5 \]
Now, let's check this value in the original inequality. Since \(47.5\) is the boundary point that results in the elevation of exactly 270 feet:
1. Check for \( x = 47.5 \):
\[
80 + 4(47.5) = 80 + 190 = 270
\]
Here, \( 80 + 4(47.5) > 270 \) is **false** since it equals 270.
2. Now check \( x = 47 \):
\[
80 + 4(47) = 80 + 188 = 268
\]
Thus, \( 80 + 4(47) > 270 \) is **false**.
3. Check \( x = 48 \):
\[
80 + 4(48) = 80 + 192 = 272
\]
Here, \( 80 + 4(48) > 270 \) is **true**, since 272 is greater than 270.
In summary:
- \( x = 47.5 \) is the boundary, yielding exactly 270 feet, which **does not satisfy** the inequality.
- \( x = 47 \) results in 268 feet, which is **not sufficient** to satisfy the inequality.
- \( x = 48 \) produces 272 feet, which **does satisfy** the inequality.
So, the solutions to the inequality \( 80 + 4x > 270 \) are \( x > 47.5 \). Therefore, Sarah can hike for any number of minutes greater than 47.5 to be above 270 feet.