Explain the following : (i) `|x|=5` (ii) `|x|=-5` (iii) `|x|lt5` (iv) `|x|lt-5` (v) `|x|gt-5` (vi) `|x|lt5`

Answer

Let's break down the explanations for each of the statements involving absolute values step by step.

### Step-by-Step Solution:

**(i) |x| = 5**

To solve the equation |x| = 5, we recognize that the absolute value of x represents the distance of x from 0 on the number line. Therefore, there are two possible cases:
1. x = 5
2. x = -5

Thus, the solutions to |x| = 5 are x = 5 and x = -5.

**(ii) |x| = -5**

The absolute value of a number is always non-negative (greater than or equal to zero). Therefore, there are no values of x that can satisfy |x| = -5.

Thus, the solution is that there are no solutions.

**(iii) |x| < 5**

The inequality |x| < 5 means that the distance of x from 0 is less than 5. This can be expressed as:
-5 < x < 5

This means x can take any value between -5 and 5, but not including -5 and 5 themselves.

**(iv) |x| < -5**

Similar to part (ii), since the absolute value is always non-negative, there are no values of x that can satisfy |x| < -5.

Thus, the solution is that there are no solutions.

**(v) |x| > -5**

Since the absolute value is always non-negative, |x| is always greater than -5 for any real number x. Therefore, this inequality is true for all real numbers.

Thus, the solution is that all real numbers are solutions.

**(vi) |x| < 5**

This is the same as part (iii). The inequality |x| < 5 means the distance of x from 0 is less than 5, which can be expressed as:
-5 < x < 5

Thus, the solutions are the same as in part (iii).

### Summary of Solutions:
1. |x| = 5 → x = 5 or x = -5
2. |x| = -5 → No solution
3. |x| < 5 → -5 < x < 5
4. |x| < -5 → No solution
5. |x| > -5 → All real numbers
6. |x| < 5 → -5 < x < 5