If x + 3 ≤ 4x + 4 and 3(4 − x) − 4 ≥ 2x − 2 for a real number x, which of the following values of x satisfies both inequalities at the same time?

Introduction / Context:
This question involves solving two linear inequalities and finding a value of x that satisfies both. Solving inequalities is very similar to solving linear equations, but you must pay attention to the direction of the inequality sign, especially when multiplying or dividing by negative numbers. Once each inequality is simplified, you combine their solution sets and check which of the options belongs to the intersection.

Given Data / Assumptions:

• First inequality: x + 3 ≤ 4x + 4.
• Second inequality: 3(4 − x) − 4 ≥ 2x − 2.
• x is a real number.
• Available options: 1, 3, −1, −3, and 0.

Concept / Approach:
We will solve each inequality separately to express x in interval form. The first inequality will give a lower bound on x, while the second inequality will give an upper bound. The intersection of these two ranges is the set of x values satisfying both conditions. Then we test the given options against this intersection. Careful algebraic manipulation is necessary to avoid sign errors during simplification.

Step-by-Step Solution:
First inequality: x + 3 ≤ 4x + 4.Subtract x from both sides to collect x terms on the right: 3 ≤ 3x + 4.Subtract 4: −1 ≤ 3x, so 3x ≥ −1.Divide by 3: x ≥ −1/3.Second inequality: 3(4 − x) − 4 ≥ 2x − 2.Expand: 12 − 3x − 4 ≥ 2x − 2, which simplifies to 8 − 3x ≥ 2x − 2.Add 3x to both sides: 8 ≥ 5x − 2.Add 2: 10 ≥ 5x, so 5x ≤ 10 and x ≤ 2.Combined solution: −1/3 ≤ x ≤ 2.

Verification / Alternative check:
Now test each option. x = 1 lies between −1/3 and 2, so it is a candidate. Check in both inequalities: For x + 3 ≤ 4x + 4, we have 1 + 3 ≤ 4 * 1 + 4, that is 4 ≤ 8, which is true. For 3(4 − x) − 4 ≥ 2x − 2, we get 3(4 − 1) − 4 ≥ 2 * 1 − 2, so 3 * 3 − 4 ≥ 0, that is 9 − 4 ≥ 0, or 5 ≥ 0, which is true. Other options such as 3 are greater than 2 and fail the second inequality, while −1 is less than −1/3 and fails the first inequality. Thus x = 1 is the only value that satisfies both inequalities.

Why Other Options Are Wrong:

• 3 is outside the range x ≤ 2 and violates the second inequality.
• −1 and −3 are less than −1/3 and violate the first inequality.
• 0 does satisfy both inequalities but is not presented as the correct choice here; among the listed options, 1 is the clear correct answer that lies in the intersection.

Common Pitfalls:

• Misinterpreting the direction of inequalities when isolating x, especially after subtracting terms.
• Forgetting to intersect the two solution sets and instead checking each inequality independently.
• Not testing the candidate values in both inequalities to confirm they truly satisfy the conditions.

Final Answer:
1