6.4: Compound Inequalities
Learning Objectives
At the end of this lesson, students will be able to:
- Write and graph compound inequalities on a number line.
- Solve a compound inequality with “and.”
- Solve a compound inequality with “or.”
- Solve compound inequalities using a graphing calculator (TI family).
- Solve real-world problems using compound inequalities.
Vocabulary
Terms introduced in this lesson:
- compound inequalities
- combined inequalities
Teaching Strategies and Tips
Use this lesson to show how two or more inequalities can be combined with “and” or “or.”
As a class, discuss the differences between “and” and "or.”
- Use real-world examples:
- Earth has one moon and Venus has no moons. (True)
- Earth has two moons and Venus has no moons. (False)
- Earth has two moons or Venus has no moons. (True)
- Earth has two moons or Venus has three moons. (False)
- Allow the discussion to lead to when “and” statements are true and when “or” statements are true.
Introduce compound inequalities in a way similar to single inequalities.
- Have students rewrite compound inequalities as individual inequalities and then solve each one separately.
- Solutions to “and” are solutions satisfying both inequalities. Solutions to “or” are solutions satisfying at least one inequality. See Example 1.
- Suggest that students graph each part of a compound inequality on two separate parallel number lines, aligned at the origin. The solution set can be graphed on a third.
Example:
Graph the following compound inequality on the number line.
\begin{align*}x \le -4 \ \ \ \text{or} \ \ \ x \ge 1.\end{align*}
Solution:
Begin by drawing three parallel number lines. On the first, graph the solution set to the first part, \begin{align*}x \le -4\end{align*}. On the second, graph the solution set to the second part, \begin{align*}x \ge 1\end{align*}. Since the statement is joined with an “or”, solutions to the compound inequality, \begin{align*}x \le -4\end{align*} or \begin{align*}x \ge 1\end{align*}, are those in at least one of the two solutions sets.
As a way to check an answer, have students choose one point from each shaded interval. Test these points in the main inequality. Also, test the endpoints to make sure the correct dot was used (solid or hollow).
Use Examples 1a, 1d, 2a, and 2d to introduce compact notation for “and.” For example, \begin{align*}x < 4\end{align*} and \begin{align*}x \ge -4\end{align*} is rewritten as \begin{align*}-4 \le x < 4\end{align*}.
Allow students in Examples 1b, 1c, 2b, and 2c to infer that a solution set consisting of two branches is “or.”
Compound inequalities with “and” can be solved without breaking them into two individual inequalities. See Example 3.
Example:
Solve the compound inequality.
\begin{align*}-1 < 2x - 3 \le 7.\end{align*}
Solution: Without rewriting the compound inequality as two separate inequalities, solve for \begin{align*}x\end{align*} by adding \begin{align*}3\end{align*} to each of the three “sides”.
\begin{align*} -1 + 3 & < 2x - 3 + 3 \le 7 + 3\\ 2 & < 2x \le 10\end{align*}
Divide each of the three “sides” by \begin{align*}2\end{align*}.
\begin{align*}\frac{2} {2} & < \frac{\cancel{2}x} {\cancel{2}} \le \frac{10} {2}\\ 1 & < x \le 5.\end{align*}
As students have trouble with “and” and “or,” encourage them to think of “and” as what’s common to two solution sets. (Ask: What’s common to this interval and that interval?) Have them think of “or” as the combination of two intervals. (Ask: What do you get when you put both intervals together?)
Check for conceptual understanding with exercises such as:
Example:
Graph the following compound inequalities on the number line.
a. \begin{align*}x < -3 \end{align*} and \begin{align*}x > 2\end{align*}
b. \begin{align*}x < -3\end{align*} and \begin{align*}x < 2\end{align*}
c. \begin{align*}x > -3\end{align*} and \begin{align*}x > 2\end{align*}
d. \begin{align*}x > -3\end{align*} and \begin{align*}x < 2\end{align*}
e. \begin{align*}x < -3\end{align*} or \begin{align*}x > 2\end{align*}
f. \begin{align*}x < -3\end{align*} or \begin{align*}x < 2\end{align*}
g. \begin{align*}x > -3\end{align*} or \begin{align*}x > 2\end{align*}
h. \begin{align*}x > -3\end{align*} or \begin{align*}x < 2\end{align*}
The exercises above exhaust all the possible cases using “and”, “or”, \begin{align*}“<”, “>”\end{align*} and two arbitrary numbers. Students see the effects of varying one characteristic at a time.
Error Troubleshooting
General Tip. Remind students to use the correct symbols:
Interval notation | Inequality notation | Solution graph | |
---|---|---|---|
Included endpoint | \begin{align*}[ , ]\end{align*} | \begin{align*}\ge , \le\end{align*} | solid dots |
Excluded endpoint | \begin{align*}( , )\end{align*} | \begin{align*}>, <\end{align*} | hollow dots |
General Tip. Encourage proper notation.
- An answer such as \begin{align*}-1 < x > -5\end{align*} is either incomplete (not simplified) or incorrect. It is unclear whether the student intended on using “and” or “or”.
- In Examples 6 and 7, discourage students from writing \begin{align*}1.875 \ge t \ge 1.56 \end{align*} and \begin{align*}8.25 \le t \le 6.75\end{align*} . Remind students to reverse the entire expression in their final answer to \begin{align*}1.56 \le t \le 1.875\end{align*} and \begin{align*}6.75 \le t \le 8.25\end{align*}.