Question

How do you graph `|16-x|>=10`?

Answer 1

See a solution process below:

Explanation:

First, we will solve the inequality. 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.

`-10 >= 16 - x >= 10`

First, subtract `color(red)(16)` from each segment of the system of inequalities to isolate the `x` term while keeping the system balanced:

`-10 - color(red)(16) >= 16 - color(red)(16) - x >= 10 - color(red)(16)`

`-26 >= 0 - x >= -6`

`-26 >= -x >= -6`

Now, multiply each segment of the system by `color(blue)(-1)` to solve for `x` while keeping the system balanced. However, because we are multiplying or dividing inequalities by a negative number we need to reverse the inequality operators:

`color(blue)(-1) xx -26 color(red)(<=) color(blue)(-1) xx -x color(red)(<=) color(blue)(-1) xx -6`

`26 color(red)(<=) x color(red)(<=) 6`

Or, `x <= 6` and `x >= 26`

Or, in interval notation:

`(-oo, 6)` and `(26, +oo)`

To graph this we will draw two vertical lines at `6` and `26` on the horizontal axis.

The lines will be a solid lines because the inequality operators contain an "or equal to" clause.

We will shade to the left and right side of the lines to display thie inequality intervals: