(a)

#### To determine

**To check:** Whether the value of *x* is less than 1 or equal to 1 if x=0.99999....

#### Answer

The value of *x* is x=1.

#### Explanation

Consider, x=0.99999....

Multiply both sides by 10.

10⋅x=10⋅(0.99999...)10x=9.9999...10x=9+0.9999...

Substitute *x* for 0.99999... in the above equation,

10x=9+x9x9=9

Divide both sides of the equation by 9.

9x9=99x=1

Therefore, the value of *x* is equal to 1. That is, x=1.

**Note:** The value of *x* is not less than one (x<1) because *x* consists of an infinite number of 9s.

(b)

#### To determine

**To find:** The value of *x*.

#### Answer

The value of *x* is x=1.

#### Explanation

**Given:**

The value of x=0.99999....

**Result used:**

The sum of the geometric series ∑n=1∞arn−1 (or) a+ar+ar2+⋯ is a1−r, where *a* is the first term of the series and *r* is the common ratio of the series.

**Calculation:**

Consider, x=0.99999... and it can be written as,

x=0.9+0.09+0.009+0.0009+⋯=910+9100+91000+910000+⋯

Clearly, it is geometric series with the first term of the series is a=910 and the common ratio of the series is

r=9100910=9100⋅109=110

Use the Result stated above, the sum of the series is

x=9101−110=91010−110=910910=1

Therefore, the value of *x* is x=1.

(c)

#### To determine

**To describe:** The decimal representations of the number 1.

#### Answer

The number 1 has two decimal representations.

#### Explanation

The decimal representations of the number 1 are 0.9999… and 1.0000….

(d)

#### To determine

**To describe:** The numbers have more than one decimal representation.

#### Answer

All terminating decimals other than zero have more than one decimal representation.

#### Explanation

**Examples:**

1. The number 2 can be written as 1.9999⋯ and 2.0000⋯.

2. The number 5.245 can be written as 5.245000⋯ and 5.244999⋯.