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:

**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.

Must Read:

- Exercise.3.1: Scientific Notation
- Exercise.3.2: Common and Natural Logarithm
- Exercise.3.3: Laws of Logarithm
- Exercise.3.4: Application of Logarithm
- Exercise.5.1: Factorization
- Exploring Various Types of Factorization
- Exercise.5.3:Remainder Theorem And Factor Theorem
- Exercise.5.4: Factorization of a Cubic Polynomial
- Exercise.6.1: Highest Common Factor
- Method of Finding Least Common Multiple
- Exercise.6.2: Basic Operations on Algebraic Fractions
- Exercise.6.3: Square Root of Algebraic Expression

**Math Full book 9th Class **

**Exercise # **7.2

**Azam Bodla**

**M.Phil. Mathematics, Content Writer, SEO ExpertWeb Developer, Online TutorCall or WhatsApp: +923059611600Gmail:azambodlaa@gmail.com**