Chapter 8.7, Problem 65HP

To determine

Explain how to solve inequalities that involve multiplication and division.

Answer to Problem 65HP

If you multiply or divide both sides of an inequality by the same negative number, the inequality reverses.

Explanation of Solution

Given:

Explain how to solve inequalities that involve multiplication and division.

Concept Used:

While solving linear equation:

When we multiply or divide by a negative to solve an equation, it's the same as multiplying or dividing by a negative to solve a regular equation.

Example:

  $\begin{array}{l}3x=81\frac{x}{3}=27\\ \frac{3x}{3}=\frac{81}{3}3·\frac{x}{3}=3·27\\ x=27x=81\end{array}$

Just remember to multiply or divide both sides by the same quantity, then simplify.

While solving linear inequality:

When we multiply or divide by a positive to solve an inequality, it's the same as multiplying or dividing by a negative to solve a regular inequality.

Example:

  $\begin{array}{l}3x<81\frac{x}{3}<27\\ \frac{3x}{3}<\frac{81}{3}3·\frac{x}{3}<3·27\\ x<27x<81\end{array}$

But, remember, when multiplying or dividing by a negative YOU MUST FLIP THE INEQUALITY SIGN.

Example:

  $\begin{array}{l}-3x<81-\frac{x}{3}<27\\ \frac{-3x}{-3}>\frac{81}{-3}-3·-\frac{x}{3}>-3·27\\ x>-27x>-81\end{array}$

The only difference is: If you multiply or divide both sides of an equation by the same negative number, the equation remains the same, but If you multiply or divide both sides of an inequality by the same negative number, the inequality reverses.

Thus, if you multiply or divide both sides of an inequality by the same negative number, the inequality reverses.