Question
Ms. Jones is hiking. She starts at 40 feet and hikes up at a constant rate of 8 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.
40+8x>270
Excellent!
You just wrote an inequality to represent the situation. Now solve the associated equation, 40 + 8x = 270, to find the boundary for the inequality. Show your work or solve on paper.
40 + 8x > 270
40+8x
=
270
x
=
28.75
Excellent!
Now figure out if the boundary, x = 28.75, is a solution to the inequality. Does x = 28.75 make the inequality true or false?
40 + 8x > 270
True
False
You got it!
You found that x= 28.75 makes the inequality false. Now test x= 28 and x= 29. Do they make the inequality true or false?
x= 28x=28.75 (boundary)x= 2940 + 8x > 270
False
True
False
False
True
True
False
Good work!
Use the table to help you write the solution to the inequality.
x= 28x=28.75 (boundary)x= 2940 + 8x > 270FalseFalseTrue
x
≤
<
≥
>
Write an inequality to represent the situation. Use x to represent the number of minutes.
40+8x>270
Excellent!
You just wrote an inequality to represent the situation. Now solve the associated equation, 40 + 8x = 270, to find the boundary for the inequality. Show your work or solve on paper.
40 + 8x > 270
40+8x
=
270
x
=
28.75
Excellent!
Now figure out if the boundary, x = 28.75, is a solution to the inequality. Does x = 28.75 make the inequality true or false?
40 + 8x > 270
True
False
You got it!
You found that x= 28.75 makes the inequality false. Now test x= 28 and x= 29. Do they make the inequality true or false?
x= 28x=28.75 (boundary)x= 2940 + 8x > 270
False
True
False
False
True
True
False
Good work!
Use the table to help you write the solution to the inequality.
x= 28x=28.75 (boundary)x= 2940 + 8x > 270FalseFalseTrue
x
≤
<
≥
>
Answers
GPT-4o mini
To summarize the work and complete the table, let’s go through the findings step-by-step and clarify the solution to the inequality.
### Step 1: Write the Inequality
The situation is represented by the inequality:
\[
40 + 8x > 270
\]
### Step 2: Solve the Associated Equation
To find the boundary:
\[
40 + 8x = 270
\]
Subtract 40 from both sides:
\[
8x = 230
\]
Divide by 8:
\[
x = 28.75
\]
### Step 3: Check if the Boundary is a Solution
Now, we will check if \( x = 28.75 \) makes the inequality true:
\[
40 + 8(28.75) = 40 + 230 = 270
\]
The inequality is \( 270 > 270 \), which is false. Thus, \( x = 28.75 \) is not a solution to the inequality.
### Step 4: Test values
- **Test \( x = 28 \)**:
\[
40 + 8(28) = 40 + 224 = 264
\]
Check: \( 264 > 270 \) → **False**
- **Test \( x = 29 \)**:
\[
40 + 8(29) = 40 + 232 = 272
\]
Check: \( 272 > 270 \) → **True**
### Step 5: Construct the Solution
Based on the values:
- For \( x = 28 \) (false) and \( x = 29 \) (true), the solution to the inequality \( 40 + 8x > 270 \) is \( x > 28.75 \).
### Final Solution
The final expression for the solution can be represented as:
\[
x > 28.75
\]
### Completing the Table
Using the findings, we can fill the table like this:
| x | 28 | 28.75 (boundary) | 29 |
|---------|-------|-------------------|---------|
| \( 40 + 8x > 270 \) | False | False | True |
### Solution Representation
From the tests, the complete solution to the inequality is:
\[ x > 28.75 \]
Therefore, Ms. Jones can hike for more than 28.75 minutes in order to be above 270 feet in elevation.
### Step 1: Write the Inequality
The situation is represented by the inequality:
\[
40 + 8x > 270
\]
### Step 2: Solve the Associated Equation
To find the boundary:
\[
40 + 8x = 270
\]
Subtract 40 from both sides:
\[
8x = 230
\]
Divide by 8:
\[
x = 28.75
\]
### Step 3: Check if the Boundary is a Solution
Now, we will check if \( x = 28.75 \) makes the inequality true:
\[
40 + 8(28.75) = 40 + 230 = 270
\]
The inequality is \( 270 > 270 \), which is false. Thus, \( x = 28.75 \) is not a solution to the inequality.
### Step 4: Test values
- **Test \( x = 28 \)**:
\[
40 + 8(28) = 40 + 224 = 264
\]
Check: \( 264 > 270 \) → **False**
- **Test \( x = 29 \)**:
\[
40 + 8(29) = 40 + 232 = 272
\]
Check: \( 272 > 270 \) → **True**
### Step 5: Construct the Solution
Based on the values:
- For \( x = 28 \) (false) and \( x = 29 \) (true), the solution to the inequality \( 40 + 8x > 270 \) is \( x > 28.75 \).
### Final Solution
The final expression for the solution can be represented as:
\[
x > 28.75
\]
### Completing the Table
Using the findings, we can fill the table like this:
| x | 28 | 28.75 (boundary) | 29 |
|---------|-------|-------------------|---------|
| \( 40 + 8x > 270 \) | False | False | True |
### Solution Representation
From the tests, the complete solution to the inequality is:
\[ x > 28.75 \]
Therefore, Ms. Jones can hike for more than 28.75 minutes in order to be above 270 feet in elevation.