Question Video: Solving Absolute Value Linear Inequalities Mathematics

Video Transcript

Find algebraically the solution set of the inequality the absolute value of eight minus 𝑥 is greater than 17.

We know that the absolute value of a function is its distance from zero. If the absolute value of eight minus 𝑥 is greater than 17, then eight minus 𝑥 must be more than 17 away from zero. This gives us two inequalities, eight minus 𝑥 is greater than 17 or eight minus 𝑥 is less than negative 17. We can solve these two inequalities by firstly adding 𝑥 to both sides. This gives us eight is greater than 17 plus 𝑥 and eight is less than negative 17 plus 𝑥.

In our first inequality, we subtract 17 from both sides, giving us negative nine is greater than 𝑥 or 𝑥 is less than negative nine. In our second inequality, we need to add 17 to both sides such that 25 is less than 𝑥 or 𝑥 is greater than 25. If we consider the number line as shown, we know that 𝑥 is less than negative nine. This means it can take any value to the left of negative nine. We also know that 𝑥 is greater than 25, so it can take any value to the right of 25.

We can therefore conclude that the solution set of the inequality the absolute value of eight minus 𝑥 is greater than 17 is the set of all real values apart from those in the closed interval between negative nine and 25. Substituting in any real value apart from those between negative nine and 25 inclusive will give a correct solution to the inequality.