Express the following symbolically: 𝑥 is greater than three and less than 18.
We recall that when dealing with inequalities, we have four possibilities. Firstly, 𝑥 is greater than some value 𝑎. Secondly, 𝑥 is greater than or equal to some value 𝑎. Thirdly, we have 𝑥 is less than 𝑎. And finally, 𝑥 is less than or equal to 𝑎. In this question, we are told that 𝑥 is greater than three and less than 18. We can write this as two separate inequalities.
Firstly, 𝑥 is greater than three. This is the same as three is less than 𝑥. Our second inequality would be 𝑥 is less than 18. This can also be written as a compound inequality as shown: 𝑥 is greater than three and less than 18.
Note that when writing a compound inequality, the inequality symbols must point in the same direction. We could also demonstrate this on a number line. As 𝑥 cannot be equal to three or 18, we have an open circle at three and 18. We then join these values with a horizontal line so that 𝑥 is greater than three and less than 18. If the statements had included the number three or the number 18, we would have colored in these circles.