Answer :
To solve the question about the range of temperatures in your area in the winter, let's break down the information and determine which inequalities apply based on the given temperatures.
### Understanding the Temperature Range:
1. Dover Winter Temperatures:
- The average low temperature in January is [tex]\(28^\circ F\)[/tex].
- The average high temperature in January is [tex]\(42^\circ F\)[/tex].
This means the temperature during winter can be any value between [tex]\(28^\circ F\)[/tex] and [tex]\(42^\circ F\)[/tex].
### Choosing Inequalities:
To represent this range using inequalities, we need conditions that reflect any temperature within this span, from the lowest to the highest.
- Choices Given:
- [tex]\(x = 28\)[/tex] and [tex]\(x = 42\)[/tex]
- [tex]\(x \leq 42\)[/tex]
- [tex]\(x \leq 28\)[/tex]
- [tex]\(42 \leq x\)[/tex]
- [tex]\(28 \leq x\)[/tex]
#### Evaluating the Choices:
1. [tex]\(x \leq 42\)[/tex]:
- This inequality states that [tex]\(x\)[/tex] can be any temperature up to [tex]\(42^\circ F\)[/tex], which is correct for our range because the maximum average temperature is [tex]\(42^\circ F\)[/tex].
2. [tex]\(x \leq 28\)[/tex]:
- This inequality limits [tex]\(x\)[/tex] to temperatures [tex]\(28^\circ F\)[/tex] or below, which does not fit our range; we expect temperatures to go above [tex]\(28^\circ F\)[/tex].
3. [tex]\(42 \leq x\)[/tex]:
- This suggests temperatures start at [tex]\(42^\circ F\)[/tex] or higher. This is not correct since [tex]\(42^\circ F\)[/tex] is the upper boundary of our range.
4. [tex]\(28 \leq x\)[/tex]:
- This states that [tex]\(x\)[/tex] can be any temperature from [tex]\(28^\circ F\)[/tex] upward, which correctly reflects the lower limit of our temperature range.
### Correct Inequalities:
Based on the choices, the correct inequalities that represent the range of temperatures from [tex]\(28^\circ F\)[/tex] to [tex]\(42^\circ F\)[/tex] are:
- [tex]\(x \leq 42\)[/tex] (temperature can go up to [tex]\(42^\circ F\)[/tex])
- [tex]\(28 \leq x\)[/tex] (temperature can start from [tex]\(28^\circ F\)[/tex])
Together, these inequalities can also be combined to explicitly express the full range as:
- [tex]\(28 \leq x \leq 42\)[/tex]
This means that the temperature, [tex]\(x\)[/tex], is at least [tex]\(28^\circ F\)[/tex] and at most [tex]\(42^\circ F\)[/tex], which fits our winter temperature data perfectly.
### Understanding the Temperature Range:
1. Dover Winter Temperatures:
- The average low temperature in January is [tex]\(28^\circ F\)[/tex].
- The average high temperature in January is [tex]\(42^\circ F\)[/tex].
This means the temperature during winter can be any value between [tex]\(28^\circ F\)[/tex] and [tex]\(42^\circ F\)[/tex].
### Choosing Inequalities:
To represent this range using inequalities, we need conditions that reflect any temperature within this span, from the lowest to the highest.
- Choices Given:
- [tex]\(x = 28\)[/tex] and [tex]\(x = 42\)[/tex]
- [tex]\(x \leq 42\)[/tex]
- [tex]\(x \leq 28\)[/tex]
- [tex]\(42 \leq x\)[/tex]
- [tex]\(28 \leq x\)[/tex]
#### Evaluating the Choices:
1. [tex]\(x \leq 42\)[/tex]:
- This inequality states that [tex]\(x\)[/tex] can be any temperature up to [tex]\(42^\circ F\)[/tex], which is correct for our range because the maximum average temperature is [tex]\(42^\circ F\)[/tex].
2. [tex]\(x \leq 28\)[/tex]:
- This inequality limits [tex]\(x\)[/tex] to temperatures [tex]\(28^\circ F\)[/tex] or below, which does not fit our range; we expect temperatures to go above [tex]\(28^\circ F\)[/tex].
3. [tex]\(42 \leq x\)[/tex]:
- This suggests temperatures start at [tex]\(42^\circ F\)[/tex] or higher. This is not correct since [tex]\(42^\circ F\)[/tex] is the upper boundary of our range.
4. [tex]\(28 \leq x\)[/tex]:
- This states that [tex]\(x\)[/tex] can be any temperature from [tex]\(28^\circ F\)[/tex] upward, which correctly reflects the lower limit of our temperature range.
### Correct Inequalities:
Based on the choices, the correct inequalities that represent the range of temperatures from [tex]\(28^\circ F\)[/tex] to [tex]\(42^\circ F\)[/tex] are:
- [tex]\(x \leq 42\)[/tex] (temperature can go up to [tex]\(42^\circ F\)[/tex])
- [tex]\(28 \leq x\)[/tex] (temperature can start from [tex]\(28^\circ F\)[/tex])
Together, these inequalities can also be combined to explicitly express the full range as:
- [tex]\(28 \leq x \leq 42\)[/tex]
This means that the temperature, [tex]\(x\)[/tex], is at least [tex]\(28^\circ F\)[/tex] and at most [tex]\(42^\circ F\)[/tex], which fits our winter temperature data perfectly.
