 Montana Council of Teachers of Mathematics # Please Ask My Dear Aunt Sally

Dear Aunt Sally,

I am solving inequalities and got to the part where we are multiplying both sides of the inequality by a negative number. I am having trouble coming up with a good explanation why the sign flips (reverses). Just because it works, is not a good reason for me. Do you have a better explanation?

Signed,
Mrs. B.

Dear Mrs. B.,

When I explain inequalities such as 2 < 4, I say that two is less than four because two is on the left side of four on the number line, therefore x < 4 means x is some number on the left side of four on the number line. When I explain multiplying by negative one I explain that when we multiply a number by negative one the result is the opposite of the original number. Opposites are the same distance from zero on the number line but on opposite sides of zero. When we multiply a number by negative one it translates the number to the opposite side of zero but remains the same distance from zero. If x < 4, then x is on the left side of four on the number line and when we multiply both sides of the inequality by negative 1, both the x and the four translate to the opposite side of zero (the same distance). If x was on the left side of four, it is now on the right side of four. Therefore the opposite of x is greater than negative four (-x > -4). This entire explanation can be made using absolute value instead of distance from zero but the first explanation is more visual for a younger group of students. Try doing this with several examples on a number line and you should see how this works. You can extend this explanation to multiplying both sides by other negative numbers.

Signed,
Your Dear Aunt Sally

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