Answer:
g) The correct option is: p/q ≤ 6
The phrase "the quotient of p and q" means p divided by q, which is expressed as p/q. The statement "the quotient of p and q is at most 6" means that the value of p/q cannot exceed 6, but it can be less than or equal to 6. Therefore, we use the less than or equal to a symbol (≤) to represent this statement.
here is an explanation of each option:
- p/q = 6: This statement indicates that the quotient of p and q is exactly 6. In other words, p/q can only be equal to 6 and not less than or greater than 6.
- p/q ≥ 6: This statement indicates that the quotient of p and q can be equal to 6 or greater than 6. In other words, p/q can be any value that is greater than or equal to 6.
- p/q > 6: This statement indicates that the quotient of p and q can be greater than 6, but not equal to 6. In other words, p/q can be any value that is greater than 6.
- p/q ≤ 6: This statement indicates that the quotient of p and q can be equal to 6 or less than 6. In other words, p/q can be any value that is less than or equal to 6.
- p/q < 6: This statement indicates that the quotient of p and q can be less than 6, but not equal to 6. In other words, p/q can be any value that is less than 6.
Here's how to express each statement as an inequality in terms of 8a^2:
p/q ≤ 6:
- Multiplying both sides by q, we get:
p ≤ 6q
- Multiplying both sides by 8a^2, we get:
8a^2p ≤ 48a^2q
- Therefore, the inequality for p/q ≤ 6 in terms of 8a^2 is:
8a^2p ≤ 48a^2q
h) The correct option is: 1/w ≤ 9
The reciprocal of a number w is 1/w. The phrase "the reciprocal of w is at least 9" means that the value of 1/w cannot be less than 9, but it can be greater than or equal to 9. Therefore, we use the less than or equal to symbol (≤) to represent this statement.
Here's an explanation of each option:
- 1/w ≥ 9: This statement indicates that the reciprocal of w is greater than or equal to 9. In other words, 1/w can be any value that is greater than or equal to 9.
- 1/w < 9: This statement indicates that the reciprocal of w is less than 9. In other words, 1/w can be any value that is less than 9.
- 1/w ≤ 9: This statement indicates that the reciprocal of w cannot be less than 9, but it can be greater than or equal to 9. In other words, 1/w can be any value that is greater than or equal to 9.
- 1/w > 9: This statement indicates that the reciprocal of w is greater than 9. In other words, 1/w can be any value that is greater than 9.
- 1/w = 9: This statement indicates that the reciprocal of w is exactly 9. In other words, 1/w can only be equal to 9 and not less than or greater than 9.
Here's how to express each statement as an inequality in terms of 8a^2:
1/w ≤ 9:
- Multiplying both sides by w, we get:
1 ≤ 9w
- Subtracting 1 from both sides, we get:
0 ≤ 9w - 1
- Multiplying both sides by 8a^2, we get:
0 ≤ 72a^2w - 8a^2
- Therefore, the inequality for 1/w ≤ 9 in terms of 8a^2 is:
0 ≤ 72a^2w - 8a^2
Note that the inequality for 1/w ≤ 9 involves a quadratic term with respect to "w," whereas the inequality for p/q ≤ 6 only involves linear terms with respect to "p" and "q."