3. The solution set for the inequality \( 3(-2 x+15)<-21 \) is determined by solving for \( x \). The solution is shown below. Step 1 Step 2 \[ \begin{aligned} 3(-2 x+15) & <-21 \\ -6 x+45 & <-21 \\ -6 x & >-66 \\ x & <11 \end{aligned} \] Step 3 Step 4 Which of the following statements describes the given solution to the inequality? A An error was made in Step 2. B An error was made in Step 3. C An error was made in Step 4. D The steps are all correct.

In the provided solution, the inequality \( 3(-2 x + 15) < -21 \) has the first step correct; however, a crucial mistake happens in Step 3. When moving the constant term from the left to the right side, the inequality should flip since the mathematical operations are applied directly to the negative coefficient of \( x \). Thus, the correct operation is to divide by -6 and flip the inequality, leading to \( x > 11 \), not \( x < 11 \). The real-world applications of inequalities are numerous! For instance, businesses often use inequalities to establish profit thresholds. By setting a profit inequality, they can determine the minimum sales needed to avoid losses. Similarly, engineers use inequalities to ensure safety margins in construction, making sure structures are built to withstand forces beyond expected limits.