6.2: Inequalities Using Multiplication and Division
Learning Objectives
At the end of this lesson, students will be able to:
- Solve an inequality using multiplication.
- Solve an inequality using division.
- Multiply or divide an inequality by a negative number.
Vocabulary
Terms introduced in this lesson:
- inequality notation
- set notation
- interval notation
- solution graph
- open and closed brackets
- including/not including
- reversing the inequality
Teaching Strategies and Tips
Use a simple example to present the four ways of expressing the solution set of an inequality:
Solve for \begin{align*}x\end{align*}:
\begin{align*} 3x & \le -24\\ \frac{\cancel{3}x} {\cancel{3}} & \le \frac{-24} {3}\\ x & \le -8\end{align*}
Solution:
- Inequality notation: \begin{align*}x \le -8\end{align*}.
- Set notation: The answer is \begin{align*}\left \{x | x \right .\end{align*} a real number, \begin{align*}\left . x \le -8 \right \}\end{align*}.
- Interval notation: \begin{align*}( - \infty, - 8 ]\end{align*}. A combination of parentheses and brackets are used. Because numbers less than \begin{align*}–8\end{align*} are solutions, negative infinity is used.
- Solution graph:
The answer is expressed as a closed circle (solid dot) at \begin{align*}-8\end{align*} and shaded to the left.
Tips on set notation:
- The vertical bar separates the variable from the condition that is used to describe the set. It is read as “such that.”
- \begin{align*}\left \{x | x \right .\end{align*} a real number, \begin{align*}\left . x \le -8 \right \}\end{align*} is read as “the set of all values of \begin{align*}x\end{align*}, such that \begin{align*}x\end{align*} is a real number less than or equal to \begin{align*}2\end{align*}.”
- Show students that set notation takes the general form: \begin{align*}\left \{ \right .\end{align*} “variable” such that “condition is true” \begin{align*}\left . \right \}\end{align*}.
- Students may not see the value of using set notation for the set \begin{align*}\left \{x | x \right .\end{align*} a real number, \begin{align*}\left . x \le -8 \right \}\end{align*} when inequality notation suffices. Suggest that although shading a number line or expressing an answer in interval notation may be easier in this example, set notation has advantages in other examples.
Discrete sets are easily described using set notation:
The number of pets belonging to students in class \begin{align*}\left \{0, 1, 2, 3, 4 \right \}\end{align*}
The solutions to an equation \begin{align*}\left \{–1, 2 \right \}\end{align*}.
The set of prime numbers \begin{align*}\left \{2, 3, 5, 7, 11, 13, 17, \ldots \right \}\end{align*}.
Tips on interval notation:
- This is the only form of solution that uses the infinity symbol.
- Point out that infinity is always paired with parentheses and never a bracket. The reason for this is because infinity is not a number and only a concept. (There is no end to the number line, so it cannot be included in the solution set.)
- The two extreme cases of interval notation are \begin{align*}(a, a)\end{align*} which represents the single number \begin{align*}a\end{align*}, and \begin{align*}(-\infty, \infty)\end{align*} which represents all real numbers.
Use Example 2 to show that inequality signs change direction when multiplying or dividing by negative numbers.
Explain why the rule is necessary.
- The number line is constructed so that the negative side is a reflection of the positive side. Multiplying or dividing a number by a negative is equivalent to reflecting it across the origin, as through a mirror.
- For any two positive real numbers on the number line, one will be further to the right than the other. Multiplying or dividing them by a negative has the effect of reflecting them across the origin. The rightmost number on the positive side of the origin becomes the leftmost number on the negative side of the origin.
- Therefore, if \begin{align*}a\end{align*} is greater than \begin{align*}b\end{align*}, then \begin{align*}-a\end{align*} is less than \begin{align*}-b\end{align*}. This means that the inequality sign changes direction when multiplying or dividing by a negative.
- In the case that one number is negative and the other positive, a simpler argument holds.
Error Troubleshooting
General Tip. Remind students to reverse the inequality sign when multiplying or dividing an inequality by a negative number.
Because the right side of the inequality in Example 2b has a fraction, suggest that students multiply both sides by \begin{align*}-1/9\end{align*} to avoid dividing the fraction.
Additional Example:
\begin{align*}-6x & < \frac{2} {9}\\ \left (-\frac{1} {6}\right ) (-6x) & < \left (- \frac{1} {6}\right ) \left (\frac{2} {9}\right ) && \text{The direction of the inequality is changed.}\\ x & > - \frac{1} {27}\end{align*}
In Examples 2b and 2d, remind students that the direction of the inequality does not change on account of the negative sign on the right side of the inequality. For example, when solving for \begin{align*}x\end{align*} in \begin{align*}12x > –30\end{align*}, do not change the direction of the inequality.