Question

Help!! Solving absolute value inequalities

Answer 1

48.
`x \geq -7` or
`x \leq 31`

51.
`x \leq 13` or
`x \geq 3`

54.
`x < (5)/(2)` or
`x > (-15)/(2)`

57.
`x < (18)/(7)` or
`x > -4`

Explanation:

48.
`| 12 - x | \leq 19`

-There are two equations:

Equation 1:

`12 - x \leq 19`

or

Equation 2:

`12 - x \geq -19`

Solving equation 1:

`12 - x \leq 19`

`12- 12 - x \leq 19 - 12`

`-x \leq 7`

`(-x)/(-1) \leq (7)/(-1)`

`x \geq -7` (Inequality sign changed, because of dividing by a negative number)

-Solving equation 2:

`12 - x \geq -19`

`12 - 12 - x \geq -19 - 12`

`-x \geq -31`

`(-x)/(-1) \geq (-31)/(-1)`

`x \leq 31` (Inequality sign changed, because of dividing by a negative number)

-Answers:

`x \geq -7` or
`x \leq 31`

51.
`| x - 8 | \leq 5`

-There are two equations:

Equation 1:

`x - 8 \leq 5`

or

Equation 2:

`x - 8 \geq - 5`

-Solving equation 1:

`x - 8 \leq 5`

`x - 8 + 8 \leq 5 + 8`

`x \leq 13`

-Solving equation 2:

`x - 8 \geq - 5`

`x - 8 + 8 \geq - 5 + 8`

`x \geq 3`

Answers:

`x \leq 13` or
`x \geq 3`

54.
`| 4x + 10 | < 20`

-There are two equations:

Equation 1:

`4x + 10 < 20`

or

Equation 2:

`4x + 10 > -20`

-Solving equation 1:

`4x + 10 < 20`

`4x + 10 - 10 < 20 - 10`

`4x < 10`

`(4x)/(4) < (10)/(4)`

`x < (5)/(2)`

-Solving equation 2:

`4x + 10 > -20`

`4x + 10 - 10 > -20 - 10`

`4x > -30`

`(4x)/(4) > (-30)/(4)`

`x > (-15)/(2)`

-Answers:

`x < (5)/(2)` or
`x > (-15)/(2)`

57.
`| 7x + 5 | < 23`

-There are two equations:

Equation 1:

`7x + 5 < 23`

or

Equation 2:

`7x + 5 > -23`

-Solving equation 1:

`7x + 5 < 23`

`7x + 5 - 5 < 23 - 5`

`7x < 18`

`(7x)/(7) < (18)/(7)`

`x < (18)/(7)`

Solving equation 2:

`7x + 5 > -23`

`7x + 5 - 5 > -23 - 5`

`7x > -28`

`(7x)/(7) > (-28)/(7)`

`x > -4`

Answers:

`x < (18)/(7)` or
`x > -4`

And we are finished.