- Simplify the inequality \(-2x - 4 \leq -16\) by adding 4 to both sides: \(-2x \leq -12\).
- Divide by \(-2\) and reverse the inequality sign to find the solution range: \(x \geq 6\).
- Check each set to see if every element satisfies \(x \geq 6\).
- The sets \(\{7, 14, 21\}\) and \(\{6, 8, 10\}\) consist entirely of values \(\geq 6\), so the final answer is \(\boxed{\{7, 14, 21\}, \{6, 8, 10\}}\).
Explanation
- Problem Analysis
The given problem requires identifying which of the provided sets of values \(\{x_1, x_2, \dots\}\) satisfy the linear inequality \(-2x - 4 \leq -16\) for every element \(x\) in the set. A set satisfies the condition if every element within it is a solution to the inequality. Our goal is to determine the range of values for \(x\) that satisfy this inequality and then check each given set against this condition.
- Solving the Inequality
To solve the inequality \(-2x - 4 \leq -16\), we isolate \(x\).
First, add \(4\) to both sides of the inequality:
\[-2x - 4 + 4 \leq -16 + 4\]
\[-2x \leq -12\]
Next, divide both sides by \(-2\). Recall that when multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed:
\[x \geq \frac{-12}{-2}\]
\[x \geq 6\]
Therefore, any value \(x\) that is greater than or equal to \(6\) is a solution to the given inequality.
- Testing the Sets
We evaluate each of the given sets by checking if every element \(x\) in the set satisfies \(x \geq 6\):
- \(\{7, 14, 21\}\): All elements are \(\geq 6\) (7 \(\geq 6\), 14 \(\geq 6\), 21 \(\geq 6\)). This set satisfies the condition.
- \(\{-6, -8, -10\}\): No elements are \(\geq 6\).
- \(\{6, 8, 10\}\): All elements are \(\geq 6\) (6 \(\geq 6\), 8 \(\geq 6\), 10 \(\geq 6\)). This set satisfies the condition.
- \(\{-1, -2, -3\}\): No elements are \(\geq 6\).
- \(\{1, 2, 3\}\): No elements are \(\geq 6\).
- \(\{-7, -14, -21\}\): No elements are \(\geq 6\).
- Conclusion
Based on our evaluation, the sets that satisfy the condition that every element \(x\) meets \(x \geq 6\) are \(\{7, 14, 21\}\) and \(\{6, 8, 10\}\). Thus, the correct sets are \(\{7, 14, 21\}\) and \(\{6, 8, 10\}\).
Examples
In engineering, design specifications often set minimum safety thresholds for components. If a component must withstand a certain pressure or temperature, you can represent this as an inequality like \(x \geq T_{min}\), where \(x\) is the operating condition of the test samples. By testing a batch of samples against this threshold, engineers can ensure that every unit in the production line meets the required quality standard. Understanding how to solve these linear inequalities allows for efficient verification of data batches against critical benchmarks.