Solve $12 x+6 \geq 9 x+12$
A. $x \geq 6$
B. $x \leq 6$
C. $x \geq 2$
D. $x \leq 2$
Answer (1)
Subtract 9 x from both sides: 3 x + 6 ≥ 12 .
Subtract 6 from both sides: 3 x ≥ 6 .
Divide both sides by 3: x ≥ 2 .
The solution to the inequality is x ≥ 2 .
Explanation
Understanding the Inequality We are given the inequality 12 x + 6 ≥ 9 x + 12 . Our goal is to isolate x on one side of the inequality to find the solution.
Isolating the x Term First, we subtract 9 x from both sides of the inequality to get the x terms on one side: 12 x + 6 − 9 x ≥ 9 x + 12 − 9 x This simplifies to: 3 x + 6 ≥ 12
Further Isolating x Next, we subtract 6 from both sides of the inequality to isolate the term with x :
3 x + 6 − 6 ≥ 12 − 6 This simplifies to: 3 x ≥ 6
Solving for x Finally, we divide both sides of the inequality by 3 to solve for x :
3 3 x ≥ 3 6 This simplifies to: x ≥ 2
Final Answer Therefore, the solution to the inequality is x ≥ 2 . This corresponds to option C.
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, when budgeting, you might want to ensure that your expenses do not exceed your income, which can be represented as an inequality. Similarly, in manufacturing, quality control often involves ensuring that a product's dimensions fall within a certain range, which can also be expressed using inequalities. Inequalities also play a vital role in optimization problems, where you aim to maximize or minimize a certain quantity subject to constraints.
