Select the correct answer.

Which inequality represents all the solutions of $-2(3 x+6) \geq 4(x+7)$?
A. $x \geq-4$
B. $x \leq-4$
C. $x \geq 8$
D. $x \leq 8$

Distribute constants on both sides of the inequality: − 6 x − 12 ≥ 4 x + 28 .
Isolate x by adding 6 x and subtracting 28 from both sides: − 40 ≥ 10 x .
Divide both sides by 10 to solve for x: − 4 ≥ x .
Express the solution: x ≤ − 4 , so the answer is x ≤ − 4 ​ .

Explanation

Analyzing the Inequality We are given the inequality − 2 ( 3 x + 6 ) ≥ 4 ( x + 7 ) . Our goal is to find the solution set for x and express it in the form x ≥ c or x ≤ c , where c is a constant. We will start by distributing the constants on both sides of the inequality.

Distributing Constants First, distribute − 2 on the left side: − 2 ( 3 x + 6 ) = − 6 x − 12 Next, distribute 4 on the right side: 4 ( x + 7 ) = 4 x + 28 So the inequality becomes: − 6 x − 12 ≥ 4 x + 28

Isolating x Now, we want to isolate x on one side of the inequality. Add 6 x to both sides: − 6 x − 12 + 6 x ≥ 4 x + 28 + 6 x − 12 ≥ 10 x + 28 Subtract 28 from both sides: − 12 − 28 ≥ 10 x + 28 − 28 − 40 ≥ 10 x

Solving for x Now, divide both sides by 10 : 10 − 40 ​ ≥ 10 10 x ​ − 4 ≥ x This is equivalent to: x ≤ − 4

Final Answer The solution to the inequality is x ≤ − 4 . Comparing this with the given options, we see that option B matches our solution.


Examples
Imagine you're managing a budget and need to ensure your expenses don't exceed a certain limit. This type of inequality helps you determine the maximum amount you can spend while staying within your budget. For example, if you have a fixed income and certain unavoidable expenses, solving an inequality can show you the upper limit on discretionary spending. This concept is also useful in various scenarios like determining safe load limits, setting performance targets, or calculating resource allocation.

Answered by GinnyAnswer | 2025-07-04