Question Video: Finding the Solution Set of an Absolute Value Inequality Algebraically | Nagwa Question Video: Finding the Solution Set of an Absolute Value Inequality Algebraically | Nagwa

# Question Video: Finding the Solution Set of an Absolute Value Inequality Algebraically Mathematics • Second Year of Secondary School

## Join Nagwa Classes

Find algebraically the solution set of the inequality |6 − 𝑥| <3

02:14

### Video Transcript

Find algebraically the solution set of the inequality the absolute value of six minus 𝑥 is less than three.

We begin by recalling that the absolute value of a number is its distance from zero on a number line. In this question, we are told that the absolute value of six minus 𝑥 is less than three, which means that six minus 𝑥 lies between negative three and three. Note that the question gives us a strict inequality, so negative three and three are not included. We can then express this as a double inequality: six minus 𝑥 is greater than negative three and less than three. Subtracting six from each part of the inequality, we have negative 𝑥 is greater than negative nine and less than negative three. We can then divide through by negative one, recalling that when multiplying or dividing by a negative number, this changes the direction of the inequality symbol. For example, a less than symbol becomes a greater than symbol.

We can therefore conclude that if the absolute value of six minus 𝑥 is less than three, then 𝑥 is greater than three and less than nine. As we are asked to give our answer using set notation, we have the open interval from three to nine. In order to check our solution, it is worth substituting a value from in our interval into the original inequality. If we let 𝑥 equals four, the left-hand side becomes the absolute value of six minus four. This is equal to the absolute value of two, which in turn is equal to two. And as this value is less than three, this suggests that our solution set of the open interval from three to nine is correct.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions