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Determine which integer will make the inequality $4x + 6 < 2x + 12$ false.


A$S:\{1\}$
B$S:\{-1\}$
C$S:\{4\}$
D$S:\{-4\}$

Verified Answer :

  • Simplify the inequality \(4x + 6 < 2x + 12\) to find the truth range: \(x < 3\).
  • Determine that the inequality is false for values where \(x \geq 3\).
  • Evaluate each option \(\{1, -1, 4, -4\}\) and find that only \(4\) satisfies \(x \geq 3\).
  • The final result is \(\boxed{4}\).

Explanation

  1. Understanding the Problem and Defining Falsity

We are given the inequality \(4x + 6 < 2x + 12\) and tasked with identifying which value from the set \(\{1, -1, 4, -4\}\) makes this statement false. An inequality is false if the original condition is not satisfied, which corresponds to the negation of the original statement. For a strict inequality \(A < B\), the negation is \(A \geq B\).

  1. Solving the Inequality

To determine the range of values for which the inequality \(4x + 6 < 2x + 12\) is true, we perform algebraic manipulation:

  1. Start with the inequality: \(4x + 6 < 2x + 12\).
  2. Subtract \(2x\) from both sides to consolidate the variable terms: \(2x + 6 < 12\).
  3. Subtract \(6\) from both sides to isolate the variable term: \(2x < 6\).
  4. Divide by \(2\): \(x < 3\).

This tells us the inequality is true for any \(x\) less than \(3\). Conversely, the inequality is false when \(x \geq 3\).

  1. Testing the Options

Now, we test each of the provided options against the "false" condition, which is \(x \geq 3\):

  1. For \(x = 1\): \(1 \geq 3\) is False. Thus, \(x=1\) makes the original inequality true.
  2. For \(x = -1\): \(-1 \geq 3\) is False. Thus, \(x=-1\) makes the original inequality true.
  3. For \(x = 4\): \(4 \geq 3\) is True. Thus, \(x=4\) makes the original inequality false.
  4. For \(x = -4\): \(-4 \geq 3\) is False. Thus, \(x=-4\) makes the original inequality true.

The only value that results in the original inequality being false is \(x = 4\).

  1. Conclusion

Based on our analysis, we have confirmed that the inequality \(4x + 6 < 2x + 12\) is true for all \(x < 3\) and false for all \(x \geq 3\). Among the given options, only \(x = 4\) satisfies the condition \(x \geq 3\), making the inequality false. The final answer is \(\boxed{4}\).

Examples

This type of analysis is crucial in computer programming and logistics. For instance, if you are designing a system where a server can handle at most 3 concurrent requests to remain efficient (i.e., you want the load \(L < 3\)), any value of \(L \geq 3\) would represent a system overload. Understanding when an inequality is false helps engineers identify failure points or boundary conditions in their systems, ensuring stability and optimal performance.

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