We start by finding the midpoint of the interval, which is enclosed by 90 and 210. We find the midpoint, or average, of these two endpoints by adding them and dividing by two:

150 is exactly 60 units away from both endpoints, 90 and 210. Since we are looking for the range of numbers that fall in between 90 and 210, this means that any possible value can't be more than 60 units away from 150. If a number is more than 60 units away from 150, in either the increasing or decreasing direction, it will be outside of the [90, 210] interval. We can express this using absolute value in the following way:

We start problems like these by finding the midpoint of the two endpoints, which in this case are 21 and 65. We find the midpoint, or average, by adding them and dividing by two:

43 is right in between 21 and 65, and is exactly 22 units away from each endpoint. Since we are looking for all numbers which fall outside of the the [21, 65] interval, we are looking for values which are further than 22 units away from 43, in both the positive and negative directions. Using absolute value, we express this as:

When you subtract 43 from any number larger than 65, the absolute value of the result will be greater than 22. Similarly, when you subtract 43 from any number less than 21, the absolute value of the result will be greater than 22.

Using the "greater than or equal to" sign is necessary in order to include the endpoints, 21 and 65, in our set of allowed ages.

Solve for positive values by ignoring the absolute value. Solve for negative values by switching the inequality and adding a negative sign to 7.

First, subtract 5 from both sides to get the absolute value expression alone.

Split this into two linear equations:

or 

The solution set is 

The absolute value gives two problems to solve. Remember to switch the "less than" to "greater than" when comparing the negative term.

or

Solve each inequality separately by adding to all sides.

or

This can be simplified to the format .

Remove the absolute value by setting the term equal to either or . Remember to flip the inequality for the negative term!

Solve each scenario independently by subtracting from both sides.

The absolute value of any number is nonnegative, so  must always be greater than . Therefore, any value of  makes this a true statement.

When solving for absolute values, remember there's an equation for positive value and another equation for negative value.

. S

ubtract  on both sides of all equations.

When solving for absolute values, remember there's an equation for positive value and another equation for negative value.

Add  to both sides of both equations.