Can a City's Budget Always Be Optimized?

📌 CONCEPT: A linear inequality is a mathematical statement that compares the value of an expression to a constant or another expression, with the use of a less-than-or-equal-to (≤) or greater-than-or-equal-to (≥) symbol.

📐 RULE / FORMULA: To solve a linear inequality, we can use the following steps: isolate the variable by performing operations that do not change the direction of the inequality, and then check the solution by plugging it back into the original inequality.

💡 WORKED EXAMPLE: Suppose we want to find the values of x that satisfy the inequality x - 2y ≤ 100. We can isolate x by adding 2y to both sides, resulting in x ≤ 100 + 2y. Then, we can substitute a value for y to find the corresponding value of x. For example, if y = 0, then x ≤ 100 + 2(0) = 100. If y = 10, then x ≤ 100 + 2(10) = 120.

⚠️ COMMON MISTAKE: Students often forget to check the solution by plugging it back into the original inequality, which can lead to an incorrect answer. Additionally, they may not isolate the variable correctly, which can result in an incorrect solution.