Verified Answer :

  • Simplify the inequality \(8 > 2k - 3\) by adding \(3\) to both sides: \(11 > 2k\).
  • Divide by \(2\) to find the range: \(k < 5.5\).
  • Check the candidate values: only \(k = 2\) is less than \(5.5\).
  • The correct option is \(\boxed{c}\).

Explanation

  1. Analyzing the Problem

We are given the inequality \(8 > 2k - 3\) and four candidate values for \(k\): \(k=8\), \(k=12\), \(k=2\), and \(k=9\). Our objective is to identify which of these values makes the inequality true. The inequality is a linear inequality, which we can solve by isolating the variable \(k\).

  1. Solving the Inequality

To solve the inequality \(8 > 2k - 3\) for \(k\), we follow algebraic steps to isolate the variable:

  1. Add \(3\) to both sides of the inequality:

\[8 + 3 > 2k - 3 + 3\]
\[11 > 2k\]

  1. Divide both sides by \(2\):

\[\frac{11}{2} > k\]
\[5.5 > k\]
Thus, the inequality is satisfied if and only if \(k\) is less than \(5.5\).

  1. Evaluating Candidates and Conclusion

Now, we test each of the provided candidate values against the condition \(k < 5.5\):

  • For \(k = 8\): \(8 < 5.5\) is false.
  • For \(k = 12\): \(12 < 5.5\) is false.
  • For \(k = 2\): \(2 < 5.5\) is true.
  • For \(k = 9\): \(9 < 5.5\) is false.

Only \(k = 2\) satisfies the condition. Therefore, option (c) is the correct answer.

Examples

Understanding and solving linear inequalities is essential in real-life resource management and logistics. For example, if you have a budget of \$800 for a project and each unit of a component costs \$200, but there's a flat delivery fee of \$300, you could model the number of units \(k\) you can afford with the inequality \(200k + 300 \leq 800\). Solving such inequalities helps in planning and decision-making by setting clear boundaries for what is feasible.

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