Exercise. 7.2: Equations Involving Absolute Value

Exercise. 7.2: Equations Involving Absolute Value: The absolute value of a number, denoted by |x|, represents its distance from zero on the number line, regardless of its sign. In other words, it tells you how far away a number is from zero, without considering whether it’s positive or negative.

Here are some key points about absolute value:

Non-negative: The absolute value of any number is always non-negative. This means it’s either positive or zero.
Positive numbers: The absolute value of a positive number is itself. For example, |5| = 5.
Negative numbers: The absolute value of a negative number is its opposite. For example, |-3| = 3.
Zero: The absolute value of zero is simply zero.Exercise. 7.2:

Here are some additional ways to understand absolute value:

Magnitude: Absolute value can be thought of as the magnitude or size of a number, ignoring its direction (positive or negative).
Distance: It represents the distance between a number and zero on the number line.

Here are some examples of absolute values:

|7| = 7 (positive number)
|-4| = 4 (negative number)
|0| = 0 (zero)

Absolute value is a fundamental concept in mathematics and has various applications in different areas, including:Exercise. 7.2:

Exercise. 7.2: Equations Involving Absolute Value

Absolute value, denoted by |x|, represents the non-negative distance of a number x from zero on the number line. Here are different examples to illustrate this concept:

1. Positive Numbers:

|5| = 5: The distance from 5 to 0 on the number line is 5 units, and since it’s positive, the absolute value is 5.

2. Negative Numbers:

|-3| = 3: The distance from -3 to 0 is 3 units, but since it’s negative, we flip the sign to get the absolute value, which is 3.

3. Zero:

|0| = 0: Zero is already at the origin, so its distance from zero is 0, and hence the absolute value is also 0.

4. Absolute Value in Expressions:Exercise. 7.2:

|7 – 2| = 5: Here, we first evaluate the expression inside the absolute value bars: 7 – 2 = 5. Then, we find the absolute value of 5, which is 5.

5. Absolute Value of Fractions and Decimals:

|1/2| = 0.5: Similar to whole numbers, we find the distance from 1/2 to 0 on the number line, which is 0.5, and that’s the absolute value.
|-2.7| = 2.7: The distance from -2.7 to 0 is 2.7, so the absolute value is 2.7.

Additional Points:Exercise. 7.2:

Remember, the absolute value is always non-negative.
Absolute value can be helpful in various applications, like finding the difference between two numbers regardless of their signs or representing magnitudes without considering direction.

Further Examples:

Distance between two points: If points A and B are at positions -2 and 5 on the number line, their absolute difference (distance) is |5 – (-2)| = 7.
Magnitude of a vector: In physics, the absolute value of a vector’s components represents its magnitude or size.

Questions & Answers

1. What is the absolute value of a number?

Answer: The absolute value of a number, denoted by |x|, represents its non-negative distance from zero on the number line.

2. How do you find the absolute value of a number?

Answer:
If the number is positive, its absolute value is itself.
If the number is negative, its absolute value is its opposite.
If the number is zero, its absolute value is also zero.

3. What are some examples of finding absolute values?

Answer:
|5| = 5
|-4| = 4
|0| = 0
|7 – 2| = 5
|1/2| = 0.5
|-2.7| = 2.7

4. What are some applications of absolute value?

Answer:
Solving equations and inequalities
Calculating distances
Representing magnitudes
Vector spaces
Calculus

5. Is the absolute value of a number ever negative?

Answer: No, the absolute value of a number is always non-negative. It can be zero or a positive number.

6. What is the difference between the absolute value and the sign of a number?

Answer: The absolute value tells you the distance from zero, regardless of the direction (positive or negative). The sign of a number tells you only whether it’s positive or negative.

7. How can absolute value be used to represent the difference between two points on a number line?

Answer: The absolute value of the difference between the coordinates of two points represents the distance between those points on the number line, regardless of the direction.

Conclusion

In conclusion, absolute value is a fundamental concept in mathematics that represents the non-negative distance of a number from zero on the number line. It helps us understand the magnitude or size of a number without considering its positive or negative sign.Exercise. 7.2:

Here are the key takeaways about absolute value:

Always non-negative: The absolute value of any number is never negative.
Positive numbers: The absolute value of a positive number is itself.
Negative numbers: The absolute value of a negative number is its opposite.
Zero: The absolute value of zero is simply zero.
Applications: Absolute value has various applications in solving equations, calculating distances, representing magnitudes, and various other mathematical fields.

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