- Divide the inequality by \(-3\), reversing the direction of the signs: \(-7 \geq x - 4 > -10\).
- Rearrange for clarity: \(-10 < x - 4 \leq -7\).
- Add \(4\) to all parts: \(-10 + 4 < x \leq -7 + 4\).
- Conclude that the solution is \(\boxed{-6 < x \leq -3}\).
Explanation
- Analyzing the Problem and Strategy
The goal is to solve the compound inequality \(21 \leq -3(x - 4) < 30\) for the variable \(x\). We will isolate \(x\) by performing algebraic operations on all three parts of the inequality. It is important to remember that when multiplying or dividing an inequality by a negative number, the inequality signs must be reversed.
- Isolating the term with x
First, we divide all three parts of the inequality by \(-3\). Since \(-3\) is negative, we reverse the inequality signs:
\[\frac{21}{-3} \geq \frac{-3(x - 4)}{-3} > \frac{30}{-3}\]
Performing the divisions gives:
\[-7 \geq x - 4 > -10\]
This can be rewritten in a more standard order as:
\[-10 < x - 4 \leq -7\]
- Solving for x and Selecting the Option
Next, we add \(4\) to all three parts of the inequality to solve for \(x\):
\[-10 + 4 < x - 4 + 4 \leq -7 + 4\]
Calculating the sums results in:
\[-6 < x \leq -3\]
Thus, the solution to the inequality is \(x \in (-6, -3]\). Matching this result with the provided options, we find it corresponds to the second option: \(-6 < x \leq -3\).
Examples
Consider a scenario where a machine operates within a specific temperature range, but its sensors are calibrated with an offset. If the temperature reading \(T\) is given by the expression \(T = -3(x - 4)\), where \(x\) is the input parameter, and the safe operating range for the machine is between \(21^\circ\text{C}\) and \(30^\circ\text{C}\), we can use this inequality to find the required input settings for \(x\). Solving this type of inequality ensures that the machine remains within its safe operating limits, preventing damage and maintaining efficiency.