What’s the mistake? Where’d it come from? How would you help?

When you’re done thinking about that, go check out where the mistake came from.

• Barry

The student doesn’t have the appropriate amount of caution for manipulating inequalities. Too many students think you treat them like equalities but write an inequality sign instead. This student has internalized the most basic rule: multiplying by a negative number is allowed as long as you reverse the inequality sign. I’d be willing to bet that this same student would do all sorts of other operations to both sides of inequalities without a second thought, because they are valid for equalities: square both sides, square root both sides, take reciprocals of both sides, etc. And although square rooting both sides is the one of these for which the rule is simple, the student probably couldn’t explain why square rooting both sides is valid.

Here, the student multiplied both sides by x+3, probably because it is second nature when dealing with equalities. The student doesn’t recognize that x+3 represents “some number”, so she should be asking herself the same question when she multiplied by -2 in the left column: “is the number negative?” Each column as written requires the extra assumption that x+3 is positive, meaning the first column should actually represent the condition x>-3. Then both columns need to be redone with the assumption x+3<0, leading to the extra restriction x<-5.

This splitting into cases is an artifact of multiplying by an unknown quantity; multiplying by a negative number like -2 is much simpler. Even if the student recognizes there is a question about validity in the work, she might not know what to do to deal with the issue. I'd definitely compare this type of work with solving absolute value equalities, where multiple cases are also considered. I would then try to drive home the point, through counterexamples and other means, that before EVERY operation on an inequality, the question needs to be asked: "Is it valid"? "Why?" Even strong students tend only to focus on the first question.