Problem 62 - Elementary Algebra

Question

A particular machine is set correctly if upon action its meter reads 0 units. One particular machine has a meter reading of -1.6 upon action. How far is this machine off its correct set ting?

Step-by-Step Solution

Answer

Answer: The machine is 1.6 units off its correct setting.

Step 1: Identify the correct setting and current reading

We are given that the machine is set correctly if upon action, its meter reads 0 units. The current reading of the machine is -1.6 units.

Step 2: Calculate the difference between the correct setting and current reading

To find the distance the machine is off its correct setting, we need to calculate the difference between the correct setting (0 units) and the current reading (-1.6 units). So, we have to find 0 - (-1.6).

Step 3: Find the absolute value of the difference

Now, we'll find the absolute value of the difference we calculated in step 2 to find the distance the machine is off its correct setting: Absolute difference = |0 - (-1.6)| = |1.6|

Step 4: State the final answer

The machine is 1.6 units off its correct setting.

Key Concepts

Real Numbers

In mathematics, **real numbers** are the set of numbers that include all possible numbers that can exist on the number line. This includes integers (like -3, 0, 5), fractions (like 1/2), and irrational numbers (like \( \sqrt{2} \) and \( \pi \)). They encompass both positive and negative numbers, as well as zero.

Real numbers are essential in measuring continuous quantities. In the context of our problem, the machine's meter reading is an example of a real number, particularly -1.6, which indicates a position on the number line. This ability to represent both whole and fractioned quantities makes real numbers very versatile for many practical applications. Understanding real numbers helps comprehend various aspects of everyday measurements and is a fundamental building block of more advanced mathematical concepts.

Negative Numbers

**Negative numbers** are numbers that are less than zero. They are often used to represent a loss, deficiency, or opposites. On a number line, negative numbers are located to the left of zero. In our machine problem, the reading of -1.6 is a negative number, indicating that the machine's setting is below the desired zero setting.

When dealing with negative numbers, there are a few key points to remember:

• Negative numbers are denoted with a minus (−) sign.
• If you subtract a negative number, you are essentially adding its absolute value. For example, 0 - (-1.6) is the same as 0 + 1.6.
• Negative numbers are used to express a variety of real-world situations, such as temperatures below freezing or elevations below sea level.

Recognizing how to properly work with negative numbers is crucial for solving problems that involve differences, such as understanding how far an object departs from its target setting.

Distance

In mathematics, **distance** refers to the measured space between two points. Distance is always a non-negative number, meaning it can't be negative. In mathematics, especially when working with numbers, we use the concept of absolute value to calculate the distance between two values. Absolute value shows how far a number is from zero on the number line, regardless of direction.

In the case of our exercise:

• The distance between the machine's reading and the correct setting is given by the absolute value of the difference.
• The formula used is \(|a - b|\) where \(a\) and \(b\) are two points on a number line. For example, \(|0 - (-1.6)| = |1.6|\).
• Here, 1.6 represents the distance the machine's meter is off its correct setting.

By understanding distance in this way, it helps provide a clear and precise measurement of deviation from a particular reference point or setting, which is vital in many practical applications.