If 2x + 4(x - 3) < -2 - x < 2x - 1, then what is the value of x that satisfies this double inequality?

Introduction / Context:
This problem involves solving a double inequality, where one expression lies between two others. Such questions test the ability to manipulate inequalities and find a range of values that satisfy both conditions at the same time. We must then check which of the given options lies in that range.

Given Data / Assumptions:

• Inequality: 2x + 4(x - 3) < -2 - x < 2x - 1.
• We need to find the value of x from the options that satisfies this compound inequality.
• x is a real number.

Concept / Approach:
A double inequality can be broken into two separate inequalities. We solve each part independently. The solution set for the compound inequality is then the intersection of the solution sets of these two parts. Finally, we check which answer option lies within this intersection. Algebraic simplification and careful handling of inequalities are crucial.

Step-by-Step Solution:
Step 1: Consider the left part of the inequality: 2x + 4(x - 3) < -2 - x. Step 2: Expand the left side: 2x + 4x - 12 = 6x - 12. Step 3: So 6x - 12 < -2 - x. Step 4: Add x to both sides: 7x - 12 < -2. Step 5: Add 12 to both sides: 7x < 10, so x < 10/7. Step 6: Now consider the right part: -2 - x < 2x - 1. Step 7: Add x to both sides: -2 < 3x - 1. Step 8: Add 1 to both sides: -1 < 3x, so x > -1/3. Step 9: Combine the two results to get -1/3 < x < 10/7. Step 10: From the options, x = 1 is the only value in the interval (-1/3, 10/7).

Verification / Alternative check:
Substitute x = 1 into the original compound inequality. Compute 2x + 4(x - 3) = 2*1 + 4(1 - 3) = 2 + 4(-2) = 2 - 8 = -6. The middle expression is -2 - x = -2 - 1 = -3. The right expression is 2x - 1 = 2*1 - 1 = 1. We see that -6 < -3 < 1, so x = 1 satisfies the double inequality.

Why Other Options Are Wrong:

• Option A: x = -1 is less than -1/3, so it does not satisfy the right side inequality.
• Option C: x = 2 is greater than 10/7, so it does not satisfy the left side inequality.
• Option D: x = -2 is far outside the valid range on the left.
• Option E: x = 0 gives -12 < -2 < -1, which fails the right comparison since -2 is not less than -1.

Common Pitfalls:
Students sometimes try to manipulate the entire compound inequality in one step and lose track of operations, or they accidentally reverse inequality signs when not required. Treating the compound inequality as two separate inequalities, solving them carefully, and finally intersecting the solution sets is the safest method.

Final Answer:
Thus, the value of x that satisfies the inequality is 1.