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


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

Verified Answer :

  • The original inequality is \(4x + 6 < 2x + 12\), which is false when \(4x + 6 \geq 2x + 12\).
  • Solving the inequality \(4x + 6 \geq 2x + 12\) yields \(2x \geq 6\), which simplifies to \(x \geq 3\).
  • Testing the provided options, only \(x = 4\) satisfies the condition \(x \geq 3\).
  • Therefore, the integer that makes the original inequality false is \(x = 4\), corresponding to the set \(\boxed{\{4\}}\).

Explanation

  1. Analyzing the Condition for Falsehood

We are given the inequality \(4x + 6 < 2x + 12\) and need to determine which integer from the given options ({1}, {-1}, {4}, {-4}) makes this inequality false. An inequality of the form \(f(x) < g(x)\) is false if its negation is true, which occurs when \(f(x) \geq g(x)\). Thus, we are looking for an integer \(x\) that satisfies the inequality \(4x + 6 \geq 2x + 12\).

  1. Solving for the Range of x Making the Inequality False

To find the values of \(x\) that make the original inequality false, we solve the inequality \(4x + 6 \geq 2x + 12\):
\[4x + 6 \geq 2x + 12\]
Subtracting \(2x\) from both sides, we get:
\[2x + 6 \geq 12\]
Next, subtracting 6 from both sides:
\[2x \geq 6\]
Dividing both sides by 2:
\[x \geq 3\]
This shows that the inequality \(4x + 6 < 2x + 12\) is false for any \(x \geq 3\).

  1. Evaluating the Options

We check each option provided against the condition \(x \geq 3\):

  • For \(x = 1\): \(1 \geq 3\) is False.
  • For \(x = -1\): \(-1 \geq 3\) is False.
  • For \(x = 4\): \(4 \geq 3\) is True.
  • For \(x = -4\): \(-4 \geq 3\) is False.

Since only \(x = 4\) satisfies the condition \(x \geq 3\), it is the integer that makes the original inequality false.

  1. Final Conclusion

Based on our analysis, the inequality \(4x + 6 < 2x + 12\) is false when \(x = 4\). Therefore, the correct set is \(S = \{4\}\).
The final answer is \(\boxed{\{4\}}\).

Examples

Imagine you are organizing a competition where a participant needs to score at least a certain number of points to move to the next round. If the requirement is defined by the inequality \(4x + 6 < 2x + 12\), you are essentially setting a condition that ensures only certain scores are allowed. By identifying when this condition fails, you can determine which scores are high enough to reach the next stage, helping you manage player progression effectively and keep the competition fair and structured.

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