Solve the following inequality?

Question

|1-10v| -2 ≥ 5

Answer 1

|1-10v| -2 ≥ 5

(1-10v)-2≥5  or  - ( 1-10v ) ≥ 5                             |a| = + a  or  - a

( 1 - 10v ) -2 ≥5------------->( 1 )

- ( 1 - 10v ) -2 ≥5---------->( 2 )

First slove the eqations ( 1 )

 ( 1 - 10v ) -2 ≥5

Add 2  to each side

 ( 1 - 10v ) -2 +2 ≥5 + 2

Simplify

( 1 - 10v ) ≥ 7

Subtract 1 from each side.

( 1 - 10v ) - 1 ≥ 7 - 1

Simplify

( - 10v ) ≥ 6

Multiply each side by negative one and flip the symbol

10v < 6

Divide each side by 10

( 10v / 10)  < ( 6/10)

Simplify

v   < 3/5

Slove the eqations ( 2 )

- ( 1 - 10v ) -2 ≥5

Multiply each side by negative one and flip the symbol

( 1 - 10v ) +2 < - 5

Subtract 2 from each side.

( 1 - 10v ) + 2 - 2 < - 5 - 2

Simplify

( 1 - 10v ) < - 7

Subtract 1 from each side.

( 1 - 10v ) -1 < - 7 -1

Simplify

( - 10 v ) < - 8

Cacel negative terms.

10 v  ≥ 8

Divide each side by 10.

( 10 v / 10) ≥  ( 8 / 10)

v ≥  4/5

there fore  4/5 < v < 3/5

Comments

Solution of the absolute inequality is  v ≤ -3/5 or v ≥ 4/5.

Answer 2

The Absolute Value Inequality is |1- 10v| -2 ≥ 5

Add 2 to each side.

Then, the inequality is |1- 10v| ≥ 7

|x| ≥ a can be written as x ≥ a or x ≤ -a

|1- 10v| ≥ 7 can be written as 1 - 10v ≥ 7 or 1- 10v ≤ -7

Solve the inequality :1 - 10v ≥ 7

- 10v ≥ 7 - 1

- 10v ≥ 6

- v ≥ 6/10

- v ≥ 3/5

v ≤ -3/5

Solve the inequality : 1 - 10v ≤ - 7

- 10v ≤ - 7 - 1

- 10v ≤ - 8

- v ≤ - 8/10

- v ≤ - 4/5

v ≥ 4/5

Therefore the solution of the absolute inequality is  v ≤ -3/5 or v ≥ 4/5

Solution set is {v∈R| v ≤ -3/5 or v ≥ 4/5}

Observe the graph , the closed circle means that -3/5 and 4/5 are the solutions of the inequality.