$\begin{array}{c}
10 n+16=6(n+2) \\
10(\square+16 \stackrel{?}{\neq 6}(\square+2)
\end{array}$

Solve the equation 10 n + 16 = 6 ( n + 2 ) by expanding and simplifying to find n = − 1 .
Analyze the inequality 10 □ + 16  = 6 ( □ + 2 ) by expanding and simplifying.
Determine that the inequality holds for all values of □ except □ = − 1 .
The solution is n = − 1 and □  = − 1 , meaning □ can be any real number except − 1 . Thus, n = − 1 , □  = − 1 ​ .

Explanation

Problem Analysis We are given an equation and an inequality involving variables n and □ respectively. Our goal is to solve the equation for n and determine the condition for the inequality to hold.

Solving the Equation First, let's solve the equation 10 n + 16 = 6 ( n + 2 ) for n . We start by expanding the right side of the equation: 10 n + 16 = 6 n + 12

Isolating the Variable Next, we subtract 6 n from both sides of the equation: 10 n − 6 n + 16 = 6 n − 6 n + 12 4 n + 16 = 12

Further Isolation Now, we subtract 16 from both sides: 4 n + 16 − 16 = 12 − 16 4 n = − 4

Solution for n Finally, we divide both sides by 4 to solve for n :
4 4 n ​ = 4 − 4 ​ n = − 1

Analyzing the Inequality Now, let's consider the inequality 10 □ + 16  = 6 ( □ + 2 ) . We expand the right side: 10 □ + 16  = 6 □ + 12

Isolating the Variable Subtract 6 □ from both sides: 10 □ − 6 □ + 16  = 6 □ − 6 □ + 12 4 □ + 16  = 12

Further Isolation Subtract 16 from both sides: 4 □ + 16 − 16  = 12 − 16 4 □  = − 4

Solution for Inequality Divide both sides by 4: 4 4 □ ​  = 4 − 4 ​ □  = − 1

Final Answer Therefore, the solution to the equation is n = − 1 , and the inequality holds for all values of □ except □ = − 1 .


Examples
Imagine you're balancing a budget where expenses and income must not be equal to avoid financial instability. Solving equations helps determine the exact balance point, while inequalities ensure you stay away from critical values that could lead to deficits. This concept applies in various scenarios, from managing personal finances to optimizing business operations, ensuring stability and preventing unwanted outcomes.

Answered by GinnyAnswer | 2025-07-04