Question

How do you evaluate `7| 10+ v | - 1\leq 90`?

Answer 1

`-23le v le3`

Explanation:

`7|10+v|-1le90`
First, we will add `1` to both sides.
`7|10+v|-1color(blue)+color(blue)1le90color(blue)+color(blue)1`
This is also equal to
`7|10+v|le91`
Now we divide both sides by `7`
`(cancel7|10+v|)/cancel7lecancel91/cancel7 13`
This means that `|10+v|le13`.
And it also means that `10-vge-13`

`10+vle13`
Subtract `10` from both sides.
`10+vcolor(blue)-color(blue)10le13color(blue)-color(blue)10`
Simplify.
`vle3`

`10+vge-13`
Subtract `10` from both sides.
`10+vcolor(blue)-color(blue)10ge-13color(blue)-color(blue)10`
Simplify.
`vge-23`

Combine it together.
`-23levle3`

Answer 2

See a solution process below:

Explanation:

First, add `color(red)(1)` to each side of the inequality to isolate the absolute value term while keeping the inequality balanced:

`7abs(10 + v) - 1 + color(red)(1) <= 90 + color(red)(1)`

`7abs(10 + v) - 0 <= 91`

`7abs(10 + v) <= 91`

Next, divide each side of the inequality by `color(red)(7)` to isolate the absolute value function while keeping the inequality balanced:

`(7abs(10 + v))/color(red)(7) <= 91/color(red)(7)`

`(color(red)(cancel(color(black)(7)))abs(10 + v))/cancel(color(red)(7)) <= 13`

`abs(10 + v) <= 13`

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

`-13 <= 10 + v <= 13`

Now, subtract `color(red)(10)` from each segment of the system of inequalities to solve for `v` while keeping the system balanced:

`-color(red)(10) - 13 <= -color(red)(10) + 10 + v <= -color(red)(10) + 13`

`-23 <= 0 + v <= 3`

`-23 <= v <= 3`

Or

`v >= -23` and `v <= 3`

Or, in interval notation:

`[-23, 3]`