Question

write an inequality using each of the following facts.

1. the fixture is designed for use with a lightbulb whose wattage does not exceed 60 watts.
2. the population of the town exceeds 250,000 this year.
3. the balance in jacks bank account is not $325.
4. pablo can spend no more than $875 for supplies.
5. the amount of time that it took to complete the wiring job is less than half of the budgeted 64 hours.
6. challenge: does the inequality \\(x \ge 0\\) have the same meaning as the statement \\(x\\) is not a negative number? explain.

Explanation:

Translate fact 6 into an inequality

Using the Translating Inequalities knowledge point
Let \(w\) represent the wattage of the lightbulb.
"Does not exceed" means less than or equal to (\(\le\)).

$$w \le 60$$

Translate fact 7 into an inequality

Using the Translating Inequalities knowledge point
Let \(p\) represent the population of the town.
"Exceeds" means strictly greater than (\(>\)).

$$p > 250000$$

Translate fact 8 into an inequality

Using the Translating Inequalities knowledge point
Let \(b\) represent the balance in Jack's bank account.
"Is not" means not equal to (\(
e\)).

$$b e 325$$

Translate fact 9 into an inequality

Using the Translating Inequalities knowledge point
Let \(s\) represent the amount Pablo can spend.
"No more than" means less than or equal to (\(\le\)).

$$s \le 875$$

Translate fact 10 into an inequality

Using the Translating Inequalities knowledge point
Let \(t\) represent the amount of time to complete the wiring job.
"Less than half of the budgeted 64 hours" means \(t < \frac{1}{2} \times 64\).

$$t < 32$$

Analyze the challenge question 11

We need to determine if \(x \ge 0\) has the same meaning as "\(x\) is not a negative number".
Real numbers are divided into three categories: positive numbers (\(x > 0\)), zero (\(x = 0\)), and negative numbers (\(x < 0\)).
If \(x\) is "not a negative number", it must be either positive or zero.
This is represented mathematically as \(x \ge 0\).
Therefore, the inequality and the statement have the exact same meaning.

Answer:

Question 6

Let \(w\) be the wattage:

$$w \le 60$$

Question 7

Let \(p\) be the population:

$$p > 250000$$

Question 8

Let \(b\) be the balance:

$$b e 325$$

Question 9

Let \(s\) be the amount spent:

$$s \le 875$$

Question 10

Let \(t\) be the time in hours:

$$t < 32$$

Question 11

Yes. Real numbers are positive, negative, or zero. If \(x\) is not negative, it must be positive or zero, which is written as \(x \ge 0\).