Inequalities: All Rules Are Not Created Equal

While working with students in several Algebra 1 classrooms this week, I noticed several prevalent misconceptions that contributed to errors in solving and interpreting inequalities. These misconceptions might be rooted in over generalizations of previously learned material instead of adapting learned concepts to new situations. Some of the misconceptions that contribute to students’ confusion regarding inequalities include:

- Treating inequalities the same as equations
- Thinking that the solution set to an inequality must be an inequality
- Lack of understanding of the symbolic meaning of inequalities

*Inequalities are the Same as Equations*

Knowing how to solve equations can help in manipulating inequalities, but saying that an inequality is the same as an equation causes problems when interpreting solutions. A student may be able to find a symbolic answer, but be unable to check whether or not an element is in the solution set. Consider an example that is often answered incorrectly:

Which of the following is a solution to the inequality 5x + 7 > 12?

A) -1 B) 0 C) 1 D) 2

I have found that many students can easily manipulate the inequality to arrive at x>1, only to select choice C, completely disregarding the inequality symbol.

Other examples require considering the structure of an expression (SMP7). In response to (x+3)(x-4)>0, students who have experienced solving (x+3)(x-4)=0, have no problem stating that (x+3)>0 and (x-4)>0, but may overlook that (x+3)<0 and (x-4)<0 will also generate solutions to the inequality. In these types of problems, creating a visual representation of the inequalities can help students determine if their answer makes sense or if it should be revised.

*The Solution Set Must Be an Inequality*

Perhaps due to a lack of exposure to different types of inequalities, I found that some students tend to overlook that solutions could take several forms, particularly when solving a system of inequalities. Consider the following systems:

A) x — 2 ≤ 4 and x — 2 ≥ 4

B) 1+ x > 4 and 3x — 6 < -12

In the first case, the solution is actually an equation (x = 6). In the second example, there is no solution since there are no numbers that satisfy both statements. In both cases, graphing the inequalities simultaneously on a number line can help students visualize the problem, trying to find where the inequalities overlap.

*Symbolic Meaning of Inequalities*

When reading x>3, one student told me that this was “x is less than 3” because she had been taught that the “pointy part points to the smaller number,” and since it was pointing towards the three, then it must be “less than.” More than other misconceptions that surfaced, this was among the most rudimentary. Not understanding the fundamental meaning of the inequality symbols also contributed to some students thinking that if 2<x<4, then the only solution must be 3. This prevented students from understanding the concept of infinitely many solutions. In this case, asking students to generate examples of inequalities in real contexts would help them construct meaning and verbally analyze solutions to inequalities.

**Addressing Misconceptions**

Inequalities play an important role in students’ understanding of equality. Often, students’ misconceptions are grounded in their misapplication of previously learned concepts. For this reason, it is important to expose students’ preconceptions before attempting to build on their prior knowledge. Additionally, when students understand inequalities visually, they are more likely to perform algebraic manipulation accurately. Using examples of inequalities applied in context helps to make learning more meaningful and sustainable. Lastly, developing conceptual understanding of the symbolic meaning of inequalities, rather than relying solely on procedural techniques and “rules” that eventually expire, might have a greater impact on students’ transfer of concepts to other topics.