- Identify the goal: isolate the variable \(x\) in the inequality \(-4 < 2x - 6 < 12\).
- Apply the inverse operation to the constant: add \(6\) to all three parts of the inequality to eliminate the \(-6\).
- The correct first step is: \(\boxed{\text{Add } 6 \text{ to all three parts.}}\)
Explanation
- Analyzing the Inequality and Determining the First Step
We have the compound inequality:
\[-4 < 2x - 6 < 12\]
To solve for \(x\), we need to isolate the variable \(x\) in the middle term. Isolating a variable usually involves performing the reverse order of operations: undoing additions or subtractions first, and then undoing multiplication or division. The term containing \(x\) is \(2x - 6\). To isolate \(2x\), we must first eliminate the \(-6\) by adding \(6\) to all parts of the inequality.
- Executing the First Step
By adding \(6\) to all parts of the inequality, we maintain the balance of the inequality and simplify the expression in the middle:
\[-4 + 6 < 2x - 6 + 6 < 12 + 6\]
This calculation results in:
\[2 < 2x < 18\]
This is the correct first step toward isolating \(x\), as it eliminates the constant term \(-6\). From this point, one would proceed by dividing all three parts by \(2\) to find the final range for \(x\).
- Conclusion
Comparing our findings with the given options, adding \(6\) to all three parts is the intended first step. The resulting inequality is \(-4 + 6 < 2x - 6 + 6 < 12 + 6\), which simplifies correctly. The other options, such as dividing by \(2\) or subtracting \(6\), would either be a later step or increase the complexity of the expression. Therefore, the first step is adding \(6\) to all three parts. The final answer is \(\boxed{\text{Add } 6 \text{ to all three parts.}}\)
Examples
In real life, suppose a laboratory technician needs to maintain a specific temperature range for a chemical reaction, defined by the expression \(-4 < 2T - 6 < 12\), where \(T\) is the temperature in degrees Celsius. To set the thermostat correctly, the technician must isolate \(T\). The first step is to adjust the baseline by adding 6 degrees, resulting in the range \(2 < 2T < 18\). This fundamental approach of isolating the variable is essential in many technical and engineering applications, such as calibrating sensors, managing financial budgets, or optimizing industrial production lines.