Two Limits

Some conditions have a lower limit and an upper limit at the same time. A room must stay between 10 and 30 degrees. A heater starts at 4 degrees and adds 2 degrees for each notch. Which notches are allowed? Call the notch $x$. The temperature at notch $x$ is $2x+4$. "Between 10 and 30" gives a chain inequality:

$$10 \le 2x+4 \le 30$$

$$6 \le 2x \le 26$$

$$3 \le x \le 13$$

Notches 3 through 13 are allowed. Two limits in the words give a chain in the algebra. You solve the chain by doing each step on all three parts.

"Between a and b" includes the ends when the story says "from a to b" or "at least a and at most b". It leaves the ends out when the story says "strictly between". Read the phrase and choose ≤ or <. Then solve the chain the same way as before.