Problem 71 - Elementary Algebra

Question

The low temperature today in Denver was \(-4^{\circ} \mathrm{F}\) and the high was \(42^{\circ} \mathrm{F}\). What is the temperature difference?

Step-by-Step Solution

Answer

Answer: The temperature difference today in Denver was 46°F.

Step 1: Identify the high and low temperatures

The high temperature today was \(42^{\circ} \mathrm{F}\) and the low temperature was \(-4^{\circ} \mathrm{F}\).

Step 2: Calculate the temperature difference

To find the difference between the high and low temperatures, subtract the low temperature from the high temperature: Temperature Difference = High Temperature - Low Temperature Temperature Difference = \(42^{\circ} \mathrm{F} - (-4^{\circ} \mathrm{F})\)

Step 3: Simplify the equation

When you subtract a negative number, it is the same as adding the positive version of that number: Temperature Difference = \(42^{\circ} \mathrm{F} + 4^{\circ} \mathrm{F}\) Temperature Difference = \(46^{\circ} \mathrm{F}\) The temperature difference today in Denver was \(46^{\circ} \mathrm{F}\).

Key Concepts

Integer Operations

Understanding integer operations is crucial for solving problems involving temperature differences. Integers are whole numbers and can be either positive, negative, or zero. In the context of temperature, these integers represent degrees of temperature. To find the difference between two temperatures, like in the exercise provided, you need to perform subtraction between integers.
Here are the steps you'll generally follow:

• Identify the integers involved (in this exercise, -4 and 42).
• Understand that to find a difference, you perform subtraction.
• Apply the operation carefully, especially when integers have different signs.

You might encounter issues when subtracting because the presence of negative numbers can complicate things. However, remembering that subtracting a negative is the same as adding the positive will help you with integer operations.

Negative Numbers

Negative numbers are essential in understanding real-world math problems, like temperature. They represent values less than zero. In temperature, a negative value indicates that it's colder than the zero point on that scale.
There are several key points to remember about negative numbers:

• They are the opposite of positive numbers and are always less than zero.
• When subtracting a negative number, you add its positive counterpart. For instance, \(42 - (-4) = 42 + 4\).
• In real-life scenarios, negative numbers can signify loss, decrease, or colder temperatures like in our exercise.

Negative numbers can sometimes be confusing, but visualizing them on a number line might help. On a number line, negative numbers extend to the left from zero, while positive numbers extend to the right.

Problem-Solving

Problem-solving in math often revolves around understanding the situation and using the right operations to find a solution. In our temperature example, we're asked to find a difference, which is a typical problem-solving task.
Follow these steps to approach similar problems:

• Read the problem carefully: Know what you are being asked to find.
• Gather the data: Identify and list the key numbers you'll need, like the high and low temperatures.
• Choose the right operation: Decide whether to add, subtract, multiply, or divide these numbers to find your answer.

By organizing your approach and understanding the numbers' roles, you'll become more efficient at problem-solving. Remember, practice with various problems is essential for mastery!