Question

Absolute Value Graph help pls?

For reference, my brackets are supposed to be absolute value bars :)

`2 <( 2x-3 ) < 7`

Solve.

Answer 1

Break it up into the two smaller inequalities:

`2 < abs(2x - 3)" " and " " abs(2x - 3) < 7`

Our solution will be all `x` that make both inequalities hold.

First inequality:

`abs(2x - 3) > 2<=>{(2x-3 > 2" "or),(2x - 3 < –2):}`

`color(white)(abs(2x - 3) > 2)<=>{(2x > 5" "or),(2x < 1):}`

`color(white)(abs(2x - 3) > 2)<=>{(x > 5/2" "or),(x < 1/2):}`

Second inequality:

`abs(2x-3) < 7<=>{(2x - 3 < 7" "and),(2x - 3 > –7):}`

`color(white)(abs(2x-3) < 7)<=>{(2x < 10" "and),(2x > –4):}`

`color(white)(abs(2x-3) < 7)<=>{(x < 5" "and),(x > –2):}`

Plotting both intervals:

`"<—––——–––—––o   o—————>"`
`"                     o————————o"`
`"<====================>"`
`"    –5 –4 –3 –2 –1  0   1   2   3   4   5   6"`

We can see they overlap on `(–2, 1/2) uu (5/2, 5)`. This is our solution:

`x in (–2, 1/2) uu (5/2, 5)`.

Below is a graph of `y = abs(2x - 3)`, marked with horizontal lines at 2 and 7. In order for `y` to be between 2 and 7, we must choose an `x` in `(–2," " 0.5)` or `(2.5," " 5)`.

graph{(y - abs(2x - 3))(y - 2)(y - 7) = 0 [-10, 10, -1, 9]}

Answer 2

Quicker method.

Explanation:

`2 < abs(2x - 3) < 7`

`<=>{(2 < +(2x - 3) < 7" "or),(2 < -(2x-3) < 7):}`

`<=>{(2 <"   " 2x - 3 < 7" "or),(2 < –2x+3 < 7):}`

`<=>{("   "5 < "   "2x < 10" "or),(–1 < –2x < "  "4):}`

`<=>{(5/2 < x < "   "5" "or),(1/2 > x > –2):}`

Solution set: `x in (–2," " 1/2) uu (5/2, " "5)`