Answer :
To solve this problem, we need to determine how many pounds of beans Caren can buy without exceeding her budget of \$10.
The inequality representing the situation is:
[tex]\[ 10 \geq 4.99 + 0.74x \][/tex]
Here are the steps to evaluate each given value of \( x \):
1. Calculate the left-hand side of the inequality for each value of \( x \):
- Substitute each value into the expression \( 4.99 + 0.74x \)
- Check if the result is less than or equal to 10.
- Also check if \( x \) is non-negative, since Caren can't buy a negative quantity of beans.
2. Classify each value based on the checks:
Given values to check: \( 0.58, 9, 7.1, -2, 6.5, 4 \)
### Evaluations:
1. For \( x = 0.58 \):
- Calculate \( 4.99 + 0.74 \times 0.58 \)
- \( = 4.99 + 0.4292 \)
- \( = 5.4192 \)
- \( 5.42 \leq 10 \) (True) and \( 0.58 \geq 0 \) (True)
- Solution to both the inequality and the situation
2. For \( x = 9 \):
- Calculate \( 4.99 + 0.74 \times 9 \)
- \( = 4.99 + 6.66 \)
- \( = 11.65 \)
- \( 11.65 \leq 10 \) (False)
- Not a solution
3. For \( x = 7.1 \):
- Calculate \( 4.99 + 0.74 \times 7.1 \)
- \( = 4.99 + 5.254 \)
- \( = 10.244 \)
- \( 10.244 \leq 10 \) (False)
- Not a solution
4. For \( x = -2 \):
- Calculate \( 4.99 + 0.74 \times (-2) \)
- \( = 4.99 - 1.48 \)
- \( = 3.51 \)
- \( 3.51 \leq 10 \) (True) but \( -2 \geq 0 \) (False)
- Solution to the inequality only
5. For \( x = 6.5 \):
- Calculate \( 4.99 + 0.74 \times 6.5 \)
- \( = 4.99 + 4.81 \)
- \( = 9.8 \)
- \( 9.8 \leq 10 \) (True) and \( 6.5 \geq 0 \) (True)
- Solution to both the inequality and the situation
6. For \( x = 4 \):
- Calculate \( 4.99 + 0.74 \times 4 \)
- \( = 4.99 + 2.96 \)
- \( = 7.95 \)
- \( 7.95 \leq 10 \) (True) and \( 4 \geq 0 \) (True)
- Solution to both the inequality and the situation
### Classification:
- Solution to both the inequality and the situation:
\(
\begin{align}
& 0.58, \\
& 6 \frac{1}{2} (6.5), \\
& 4
\end{align}
\)
- Solution to the inequality only:
\(
\begin{align}
& -2
\end{align}
\)
- Not a solution:
\(
\begin{align}
& 9, \\
& 7.1
\end{align}
\)
So, the final arrangement should look like this:
Solution to both the inequality and the situation:
\( 0.58, 6.5, 4 \)
Solution to the inequality only:
\( -2 \)
Not a solution:
[tex]\( 9, 7.1 \)[/tex]
The inequality representing the situation is:
[tex]\[ 10 \geq 4.99 + 0.74x \][/tex]
Here are the steps to evaluate each given value of \( x \):
1. Calculate the left-hand side of the inequality for each value of \( x \):
- Substitute each value into the expression \( 4.99 + 0.74x \)
- Check if the result is less than or equal to 10.
- Also check if \( x \) is non-negative, since Caren can't buy a negative quantity of beans.
2. Classify each value based on the checks:
Given values to check: \( 0.58, 9, 7.1, -2, 6.5, 4 \)
### Evaluations:
1. For \( x = 0.58 \):
- Calculate \( 4.99 + 0.74 \times 0.58 \)
- \( = 4.99 + 0.4292 \)
- \( = 5.4192 \)
- \( 5.42 \leq 10 \) (True) and \( 0.58 \geq 0 \) (True)
- Solution to both the inequality and the situation
2. For \( x = 9 \):
- Calculate \( 4.99 + 0.74 \times 9 \)
- \( = 4.99 + 6.66 \)
- \( = 11.65 \)
- \( 11.65 \leq 10 \) (False)
- Not a solution
3. For \( x = 7.1 \):
- Calculate \( 4.99 + 0.74 \times 7.1 \)
- \( = 4.99 + 5.254 \)
- \( = 10.244 \)
- \( 10.244 \leq 10 \) (False)
- Not a solution
4. For \( x = -2 \):
- Calculate \( 4.99 + 0.74 \times (-2) \)
- \( = 4.99 - 1.48 \)
- \( = 3.51 \)
- \( 3.51 \leq 10 \) (True) but \( -2 \geq 0 \) (False)
- Solution to the inequality only
5. For \( x = 6.5 \):
- Calculate \( 4.99 + 0.74 \times 6.5 \)
- \( = 4.99 + 4.81 \)
- \( = 9.8 \)
- \( 9.8 \leq 10 \) (True) and \( 6.5 \geq 0 \) (True)
- Solution to both the inequality and the situation
6. For \( x = 4 \):
- Calculate \( 4.99 + 0.74 \times 4 \)
- \( = 4.99 + 2.96 \)
- \( = 7.95 \)
- \( 7.95 \leq 10 \) (True) and \( 4 \geq 0 \) (True)
- Solution to both the inequality and the situation
### Classification:
- Solution to both the inequality and the situation:
\(
\begin{align}
& 0.58, \\
& 6 \frac{1}{2} (6.5), \\
& 4
\end{align}
\)
- Solution to the inequality only:
\(
\begin{align}
& -2
\end{align}
\)
- Not a solution:
\(
\begin{align}
& 9, \\
& 7.1
\end{align}
\)
So, the final arrangement should look like this:
Solution to both the inequality and the situation:
\( 0.58, 6.5, 4 \)
Solution to the inequality only:
\( -2 \)
Not a solution:
[tex]\( 9, 7.1 \)[/tex]
