9.7 Writing Inequalities
In this unit, there have been three occasions when checks have been made to see whether the result is greater than or less than some other value. The first case was in calculating the BMI and determining whether the person was overweight or underweight; the second case was in determining from footprints whether a person was walking or running; and the third case was in checking whether a phone had been used for more than 30 minutes. Checking whether values are greater than or less than some limit happens frequently, particularly with safety issues, but elsewhere, too. For example, medicines may have to be stored at a temperature of 77°F or less. Child train tickets can be bought for children who are 5 or more years old but less than 16 years old.
Rather than writing out “greater than” or “less than,” shorthand notation is often used as shown below.
[ If you have difficulty interpreting these symbols, you can think of them as arrows that point to the smaller number or remember that < looks like an L, which stands for less than. ]
> greater than
≥ greater than or equal to
< less than
≤ less than or equal to
The symbols can be read from left to right. For example, 11 > 9 is read as “11 is greater than 9” and vacation cost in dollars < 1,000 is read as “vacation cost is less than $1,000.”
To use the symbols in your own writing, decide what you want to say first, then use the symbol. For example, since 10 is greater than 5, this could be written as 10 > 5 or since on the number line, lies to the left of , is less than , and this could be written as < .
Similarly, the instructions for the medicine could be written as: Medicine temperature in °F ≤ 77.
Sometimes, you may find it helpful to draw a number line to visualize this kind of information. For example, the ages which children are eligible for the child train fare are from their fifth birthday up to but not including their 16th birthday. This means that the age has to be greater than or equal to 5 and less than 16. [ Note how drawing a diagram here helps you to see what is happening. ]
The empty circle means that this number (16) is not included and the filled-in circle means that this number (5) is included in the interval. This can be written as: 5 ≤ age for child train fare < 16.
Note carefully the format in which this last inequality is written. The variable that is being described, that is the age for a child train fare, is always in the center when defining a range of values with an upper and lower limit as we have here. This is the math convention that we follow with ranges of values.