Let's analyze the given information to determine the most appropriate inequality for the weight limit problem.
Given Information:
1. Let [tex]\( x \)[/tex] represent the number of math books.
2. Let [tex]\( y \)[/tex] represent the number of workbooks.
3. Each math book weighs 3.2 pounds.
4. Each workbook weighs 9.8 pounds.
5. The total weight of all books ordered should not exceed 50 pounds.
Translate the Information into a Mathematical Inequality:
To find out the total weight of the books ordered, we can write the total weight as the sum of the weight of the math books and the weight of the workbooks. Mathematically, this translates to:
[tex]\[ \text{Total weight} = (3.2 \, \text{pounds}) \times x + (9.8 \, \text{pounds}) \times y \][/tex]
Given that the total weight cannot be more than 50 pounds, we set up the inequality:
[tex]\[ 3.2x + 9.8y \leq 50 \][/tex]
Let's evaluate the given options:
1. [tex]\( 3.2x + 0.8y < 50 \)[/tex]:
- This inequality is incorrect because it incorrectly states the coefficient of [tex]\( y \)[/tex] as 0.8 instead of 9.8, and it uses a strict inequality (<), which is not what we have deduced.
2. [tex]\( 3.2x + 9.8y \leq 50 \)[/tex]:
- This inequality correctly reflects the weight considerations given in the problem and matches the derived inequality. It accounts for the sum of the weights of math books and workbooks and correctly uses the 'less than or equal to' (≤) sign.
3. [tex]\( 0.8x + 3.2y = 50 \)[/tex]:
- This equation is incorrect because the coefficients of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are incorrect, and it represents an equation rather than an inequality. It also wrongly states an equality which does not reflect the 'not more than 50 pounds' condition.
4. [tex]\( 0.8x + 32y \leq 50 \)[/tex]:
- This inequality is incorrect because it contains incorrect coefficients for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and does not match with our given problem requirements.
Therefore, the correct inequality is:
[tex]\[ 3.2x + 9.8y \leq 50 \][/tex]
This inequality accurately represents the weight constraint for the number of math books [tex]\( x \)[/tex] and workbooks [tex]\( y \)[/tex] that Mr. Ishenoco ordered for his class.
