Question

How do you solve `abs(x-10)>=20`?

Answer 1

Use the piecewise definition: `|f(x)|={(f(x);f(x)>=0),(-f(x);x<0):}` to separate into two inequalities.
Solve the two inequalities.
Check.

Explanation:

Given: `|x-10|>=20`

Use the piecewise definition: `|f(x)|={(f(x);f(x)>=0),(-f(x);x<0):}` to separate into two inequalities.

`x-10>=20` and `-(x-10)>=20`

Multiply both sides of the second inequality by -1:

`x-10>=20` and `x-10<=-20`

Add 10 to both sides of both inequalities:

`x>=30` and `x<=-10`

Check the equality points:

`|30-10|>=20` and `|-10-10|>=20`

`20 = 20 and 20 = 20`

Check 31 and -11:

`|31-10|>=20` and `|-11-10|>=20`

`21>=20 and 21 >= 20`

This checks.

Answer 2

Absolute value inequality. Okay. Here we go.

Explanation:

We have:

`|x-10| >= 20`

The absolute value of `x-10` can be either negative or positive. We don't know which, because we don't know yet what `x` is, so:

So either:

`x-10 >= 20`, or

`-(x-10) >= 20`.

(Just like `|x|>=20` would be: either `x>=20` or `-x>=20`. Take the whole thing inside the absolute value brackets.)

First let's solve `x-10 >= 20`:

Add `10` to both sides to isolate `x`.

We get `x >= 30`.

Next let's solve `-(x-10) >= 20`:

Multiply the `-1` into the parentheses:

`-x+10 >= 20`

Subtract `10` from both sides:

`-x>=10`.

At this point, we need to multiply both sides of the inequality by `-1` in order to solve for `x`.

When you multiply or divide an inequality by a negative number , you have to change the direction of the inequality , so we get:

`x<=-10` for this part.

The solution to this inequality is in two parts:

`x<=-10` OR `x >= 30`.

You can write this in interval notation as:

`(-oo,-10] uu [30, +oo)`.

Parentheses: that endpoint is not included in the interval - when you have only `>` or `<`.

Square brackets: that endpoint is included in the interval - when you have `>=` or `<=`.

`-oo` and `+oo` are never included in an interval because they are not fixed values. (Exactly where is infinity?)

`OR` is a logical operator that means the Union of two sets - the two parts - written `uu`.

Here's a line graph of the solution:

`-oo`<*****]---------------|------------------------[*****>`+oo`
`.....................-10.............0......................30..................`

The dots between the `-10`, the `0` and the `30` are only to keep these numbers below the correct place on the line graph.

The asterisks are shaded parts of the line to indicate that this part of the Real Number Line is part of the solution to the inequality.

I'll come back and edit this answer if I can get a better graph going.
Connie