Question

Help!! Solving absolute value inequalities

Answer 1

47.
`x < 7` or
`x > -9`

50.
`x > 7` or
`x < 17`

53.
`x < (-4)/(3)` or
`x > (32)/(3)`

56.
`x > -28` or
`x < -52`

Explanation:

47.
`| x + 1| < 8`

-There are two equations:

Equation 1:

`x + 1 < 8`

or

Equation 2:

`x + 1 > -8`

-Solving equation 1:

`x + 1 < 8`

`x + 1 - 1< 8 - 1`

`x < 7`

-Solving equation 2:

`x + 1 > -8`

`x + 1 - 1 > -8 - 1`

`x > -9`

Answers:

`x < 7` or
`x > -9`

50.
`| x + 5 | > 12`

-There are two equations:

Equation 1:

`x + 5 > 12`

or

Equation 2:

`x + 5 < -12`

-Solving equation 1:

`x + 5 > 12`

`x + 5 - 5 > 12 - 5`

`x > 7`

-Solving equation 2:

`x + 5 < -12`

`x + 5 - 5 < -12 - 5`

`x < -17`

-Answers:

`x > 7` or
`x < -17`

53.
`| 14 - 3x | > 18`

-There are two equations:

Equation 1:

`14 - 3x > 18`

or

Equations 2:

`14 - 3x < -18`

-Solving equation 1:

`14 - 3x > 18`

`14 - 14 - 3x > 18 -14`

`-3x > 4`

`(-3x)/(-3) > (4)/(-3)`

`x < (-4)/(3)` (Inequality sign changed, because of dividing by a negative number)

-Solve equation 2:

`14 - 3x < -18`

`14 - 14 - 3x < -18 -14`

`-3x < -32`

`(-3x)/(-3) < (-32)/(-3)`

`x > (32)/(3)` (Inequality sign changed, because of dividing by a negative number)

-Answers:

`x < (-4)/(3)` or
`x > (32)/(3)`

56.
`| 20 + (1)/(2)x | > 6`

-Switch the equation:

`| (1)/(2)x + 20 | > 6`

-There are two equations:

Equations 1:

`(1)/(2)x + 20 > 6`

or

Equation 2:

`(1)/(2)x + 20 < -6`

-Solving equation 1:

`(1)/(2)x + 20 > 6`

`(1)/(2)x + 20 - 20 > 6 - 20`

`(1)/(2)x > -14`

`2 * (1)/(2)x > 2 * -14`

`x > -28`

-Solving equation 2:

`(1)/(2)x + 20 < -6`

`(1)/(2)x + 20 - 20 < -6 - 20`

`(1)/(2)x < -26`

`2 * (1)/(2)x < 2 * -26`

`x < -52`

-Answers:

`x > -28` or
`x < -52`

And were done.