Verified Answer :

  • Recognize that the inequality \(4x + 6 < 2x + 12\) is false when \(4x + 6 \geq 2x + 12\).
  • Simplify the condition to \(2x \geq 6\), which leads to \(x \geq 3\).
  • Test each given option to see which satisfies \(x \geq 3\).
  • The value \(x=4\) satisfies \(x \geq 3\), making the original inequality false, so the answer is \(\boxed{4}\).

Explanation

  1. Analysis of the Inequality

We are given the inequality \(4x + 6 < 2x + 12\). We need to determine which integer from the provided set \(\{1, -1, 4, -4\}\) makes this inequality false. An inequality is false when the opposite relation holds. Specifically, the statement \(f(x) < g(x)\) is false when \(f(x) \geq g(x)\). Therefore, we need to find which of the given values satisfies \(4x + 6 \geq 2x + 12\).

  1. Solving the Inequality for the False Condition

To solve the inequality \(4x + 6 \geq 2x + 12\), we isolate \(x\) on one side:
Subtract \(2x\) from both sides:
\(4x - 2x + 6 \geq 12\)
\(2x + 6 \geq 12\)
Subtract 6 from both sides:
\(2x \geq 6\)
Divide by 2:
\(x \geq 3\)
So, the inequality \(4x + 6 < 2x + 12\) is false for any integer \(x\) that is greater than or equal to 3.

  1. Testing the Provided Options

Now we evaluate each of the given options 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 the inequality \(4x + 6 < 2x + 12\) is false when \(x \geq 3\), and \(x = 4\) is the only option that satisfies this condition, it is the correct choice. We can double-check: \(4(4) + 6 = 22\) and \(2(4) + 12 = 20\). Since \(22\) is not less than \(20\), the original inequality is indeed false for \(x = 4\).

Examples

Understanding how to identify when an inequality fails is essential in real-world scenarios such as resource management or budget planning. For instance, if you have a budget constraint, you might model your costs as an inequality. Identifying the value that makes your budget constraint false (i.e., you exceed your budget) helps you adjust your spending plans to remain within limits. This logic is fundamental for identifying thresholds and ensuring stability in various systems.

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