Video: Solving Absolute Value Inequalities

Find the solution set of the inequality |𝑥 − 6| ≥ 7.

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Video Transcript

Find the solution set of the inequality the absolute value of 𝑥 minus six is greater than or equal to seven.

In order to solve any absolute value problem of this type, we need to solve two inequalities. This is because the absolute value of a number is its distance from zero. In this question, we want the distance to be greater than or equal to seven. Firstly, the value inside our absolute value, 𝑥 minus six, could be greater than or equal to seven. Alternatively, 𝑥 minus six could be less than or equal to negative seven. Adding six to both sides of our first inequality gives us 𝑥 is greater than or equal to 13, and adding six to both sides of our second inequality gives us 𝑥 is less than or equal to negative one.

We can demonstrate this on a number line. As 𝑥 is greater than or equal to 13, we have a closed circle at 13 and 𝑥 can take any value to the right of this. Likewise, as 𝑥 is less than or equal to negative one, 𝑥 can take any value including and to the left of this value.

This means that the solution set contains all real values apart from those between negative one and 13. This can be written as the real values minus the open interval from negative one to 13.

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