Question

solve.

1. julio bought a new watt/var transducer for \\$957.50. he paid \\$145 down and will pay \\$32.50 a month until it is fully paid for.

(a) write an inequality to describe how many months it will take before he has fully paid for the transducer.
(b) how many months will it take before he has fully paid for the transducer?

1. four times the sum of a number and 3 is less than 28. what numbers satisfy this condition?
2. challenge: the perimeter of a square is greater than 64 inches but less than 152 inches. what are the possible lengths for the side of the square?

Explanation:

Define variables and model Question 17(a)

Using the Translating Word Problems and Algebraic Modeling knowledge points, let \(m\) represent the number of months.
The total amount paid consists of a down payment of \(\$145\) plus monthly payments of \(\$32.50\) per month:

$$145 + 32.50m$$

To fully pay for the transducer, this total paid amount must be greater than or equal to the total cost of \(\$957.50\):

$$145 + 32.50m \ge 957.50$$

Solve the inequality for Question 17(b)

Using the Solving Linear Inequalities knowledge point, we solve for \(m\):

$$
LATEXBLOCK0
$$

It will take at least 25 months.

Translate and solve Question 18

Using the Translating Inequalities and Solving Linear Inequalities knowledge points, let \(x\) be the number.
"Four times the sum of a number and 3" is written as:

$$4(x + 3)$$

This expression is "less than 28":

$$
LATEXBLOCK1
$$

The numbers that satisfy this condition are all real numbers less than 4.

Translate and solve Question 19

Let \(s\) represent the side length of the square.
The perimeter of a square with side length \(s\) is \(4s\).
The perimeter is greater than 64 inches but less than 152 inches:

$$64 < 4s < 152$$

Divide all parts of the compound inequality by 4:

$$
LATEXBLOCK2
$$

The possible side lengths are between 16 inches and 38 inches.

Answer:

Question 17

(a) \(145 + 32.50m \ge 957.50\) (where \(m\) is the number of months)
(b) At least 25 months

Question 18

\(x < 4\) (all real numbers less than 4)

Question 19

\(16 < s < 38\) (any side length between 16 inches and 38 inches)