17. Algebraic expression: Total cost = 2(16 + 10 + x) = 32 + 20 + 2x = 52 + 2x
18. No, the two expressions are not equivalent.
19. The cost of 3 litter boxes is equal to the cost of 12 cat toys or 4 bags of cat food, but it is not less than either of them. Therefore, the cost of 3 litter boxes is the most.
20. Distributive Property allows us to distribute the coefficient 2 to each of the terms in the expression ℓ + w.
21. The expression 2x + 2x can only be used to calculate the perimeter of a square, which is a special type of rectangle.
22. The two expressions are equivalent.
23. The two expressions are equivalent
24. The expression that is equivalent to (4_{2}^{1}+(3t+t_{2}^{1}).)
Question 17: a. Ms. Martinez bought x number of litter boxes and 8 bags of cat food for the animal shelter.
Algebraic expression: Total cost = 8 + x
b. Two sisters each bought 1 litter box, 10 cat toys, and x bags of cat food.
Algebraic expression: Total cost = 2(16 + 10 + x) = 32 + 20 + 2x = 52 + 2x
Question 18:Suppose that x has the same value in both of the expressions you wrote in Exercise 17.
Are the two expressions you wrote equivalent? Explain.
Answer: No, the two expressions are not equivalent.
Even if x has the same value in both expressions, the cost of Ms. Martinez's purchase will always be less than the cost of the two sisters' purchase, because they are buying more items.
Question 19:Communicate and Justify Which costs the most: 12 cat toys, 4 bags of cat food, or 3 litter boxes? Explain.
Answer:The cost of 3 litter boxes is the most. We can compare the costs by using the following algebraic expressions:
Cost of 12 cat toys = 12 * $4 = $48
Cost of 4 bags of cat food = 4 * $12 = $48
Cost of 3 litter boxes = 3 * $16 = $48
As we can see, the cost of 3 litter boxes is equal to the cost of 12 cat toys or 4 bags of cat food, but it is not less than either of them.
Therefore, the cost of 3 litter boxes is the most.
Question 20: Choose Efficient Methods The formula for the perimeter of a rectangle is 2ℓ + 2w, where ℓ is the length and w is the width.
How can you use the Distributive Property to write an equivalent expression for 2ℓ + 2w?
Answer: We can use the Distributive Property to write an equivalent expression for 2ℓ + 2w as follows: 2ℓ + 2w = 2(ℓ + w)
This is because the Distributive Property allows us to distribute the coefficient 2 to each of the terms in the expression ℓ + w.
Question 21: Higher Order Thinking Explain why the expression you wrote in Exercise 20 may be easier to use than (2x+2x).
Answer: The expression 2(ℓ + w) may be easier to use than 2x + 2x because it is more general.
The expression 2(ℓ + w) can be used to calculate the perimeter of any rectangle, regardless of its dimensions.
However, the expression 2x + 2x can only be used to calculate the perimeter of a square, which is a special type of rectangle.
Question 22: Communicate and Justify Zach says that the expressions 6x-35 and 3(2x-12) are equivalent because of the Distributive Property. Do you agree? Explain.
Answer: Yes, the expressions 6x-35 and 3(2x-12) are equivalent because of the Distributive Property.
We can use the Distributive Property to expand the expression 3(2x-12) as follows:
3(2x-12) = 3 * 2x - 3 * 12
= 6x - 36
Therefore, the two expressions are equivalent.
Question 23: Are the two expressions shown below equivalent? Explain.
-4n-6m-12k and \(-2(2n+3m-6k)\)
Answer: Yes, the two expressions are equivalent.
We can use the Distributive Property to expand the expression -2(2n+3m-6k) as follows:
-2(2n+3m-6k) = -2 * 2n - 2 * 3m + 2 * 6k
= -4n - 6m + 12k
Therefore, the two expressions are equivalent.
Question 24:
Select each expression that is equivalent to (4_{2}^{1}+(3t+t_{2}^{1}).)
Answers:(4_{2}^{1}+3t)+1\frac{1}{2}3(2+t).