Question Video: Solving Absolute Value Linear Inequalities Mathematics

Video Transcript

Find algebraically the solution set of the inequality the absolute value of seven minus 𝑥 plus three is less than or equal to negative six.

When we look at our inequality, we have a term with absolute value. And that means our first step is going to be to isolate this absolute value. We can do that by subtracting three from both sides of our inequality. And then, we have the absolute value of seven minus 𝑥 is less than or equal to negative nine.

At this point, we should think really carefully about this inequality. What kind of values will the absolute value of seven minus 𝑥 produce? They will produce values that are not negative, nonnegative values. This means we’ll either get a positive result or zero. But the absolute value of seven minus 𝑥 can never produce a negative value.

On the right-hand side of the inequality, we have the constant negative nine. Is there ever going to be a time where a positive number or a zero is a less than or equal to negative nine? And the answer is no. This means there is no value for 𝑥 that can make this inequality true, and there is therefore no solution. However, we want to write this in set notation, and so we write it as the empty set.

The solution set of this inequality is the empty set, as there is no solution.