Express the following as an absolute value inequality: 𝑥 is greater than four, 𝑥 is less than negative four.
We can think about these values of 𝑥 on a number line. We’ll sketch a number line with negative four, zero, and four. 𝑥 cannot be equal to four; it must be greater than four. And for our notation here, that means we will not fill in this circle. And then we’ll draw a line with an arrow to the right because 𝑥 can be equal to all the values larger than four. And we’ll draw the same kind of line for 𝑥 is less than negative four.
We often say that we measure the absolute value as the distance from zero for some value. But if we start at zero, we are saying that the distance from zero to 𝑥 must be greater than four units. We can use any value for 𝑥 so long as it is more than four units away from zero in either direction. And that means the absolute value of 𝑥 must be greater than four. As long as the distance from 𝑥 to zero is larger than four, it’s in our range.