Multiplying and dividing: reversing the sign

Multiplying or dividing both sides of an inequality by a positive number keeps the inequality sign. Multiplying or dividing both sides by a negative number reverses the sign: \(<\) becomes \(>\), and \(\le\) becomes \(\ge\).

Why: dividing \(4<10\) by \(2\) gives \(2<5\), and \(2\) lies to the left of \(5\). Multiplying by \(-1\) gives \(-2\) and \(-5\), which are the mirror images of \(2\) and \(5\) in \(0\). The mirror swaps left and right, so \(-2\) now lies to the right of \(-5\) and \(-2>-5\). Every multiplication or division by a negative number includes this reflection.

Method

1. Divide both sides by the coefficient of the unknown (or multiply by its reciprocal).
2. If that number is negative, reverse the inequality sign in the same step. If it is positive, keep the sign, even when the other side is negative.
3. Simplify the boundary value, leaving it as a fraction or mixed number if it is not a whole number.
4. Represent the solution on a number line and check with one value.

Example

Solve \(-4x\ge10\).

\(\begin{aligned} -4x&\ge10 \\ x&\le\dfrac{10}{-4} \\ x&\le-2\tfrac{1}{2} \end{aligned}\)

Dividing by \(-4\) reverses \(\ge\) to \(\le\).

Check: \(x=-3\) gives \(-4(-3)=12\), and \(12\ge10\) is true. A value such as \(x=0\), on the other side, gives \(0\ge10\), which is false.

Check: reverse only when you multiply or divide by a negative number. For \(5x<-15\), divide by \(5\): \(x<-3\). The negative number on the right does not reverse the sign.