Question

How do you solve `abs(x-3)>7`?

Answer 1

`x<-4,x>10`

Explanation:

When working with absolute value problems, we need to keep in mind that while the value of the term inside the absolute value might be negative, the absolute value will always return a positive value. Because of this, we need to evaluate both the positive and negative possibility:

Positive

`abs(x-3)>7`

`x-3>7`

`x>10`

Negative

`abs(x-3)>7`

`-(x-3)>7`

`-x+3>7`

`-x>4`

(Remember that when multiplying or dividing by a negative number on both sides of an equality, we need to flip flop the inequality sign):

`x<-4`

Putting it together

We now have the values that `x` can be:

`color(blue)(ul(bar(abs(color(black)(x<-4,x>10))))`

To graph this, place hollow dots around the numbers `-4` and `10` on a number line (the hollow line indicates that `-4` and `10` are not part of the solution), and then draw rays heading to the left (more negative) from the `-4` and to the right (more positive) from the 10.