Chapter 1.5, Problem 55PPSE

Solution

The error in the given calculation.

The classmate should not have changed the inequality sign in the final step as the inequality is not multiplied or divided by any negative number.

Given:

A classmate solved the inequality $\frac{1}{2}\left(y-16\right)\ge y+2$ as shown below

  

Concept Used:

• If there is the strict inequality symbol $<\text{ or}>$, then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$, then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$, use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$, use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.
• Average is the quotient of the sum of all data points and the number of data points.

Calculation:

To check the answer of classmate, substitute $y=-4$ into the given inequality and simplify further as shown below

  $\begin{array}{c}\frac{1}{2}\left(y-16\right)\ge y+2\\ \frac{1}{2}\left(-4-16\right)\ge -4+2\\ \frac{1}{2}\left(-20\right)\ge -2\\ -10\ge -2\text{False statement}\end{array}$

It implies that the solution of the classmate is incorrect.

The classmate should not have changed the inequality sign in the final step as the inequality is not multiplied or divided by any negative number.

Thus, the correct solution to the inequality should be $y\le -20$ .