Consider the compound inequality 5x − 4 ≤ 2 − x and 4x + 5 > 2x − 5. Determine which of the given values of x satisfies both inequalities simultaneously.

Introduction / Context:
This question assesses your ability to solve and interpret compound linear inequalities. Instead of solving only one inequality, you have to find values of x that satisfy two conditions at the same time. Such problems are common in aptitude tests, as they combine algebraic manipulation with logical reasoning about intervals on the number line.

Given Data / Assumptions:

• The first inequality is 5x − 4 ≤ 2 − x.
• The second inequality is 4x + 5 > 2x − 5.
• x is a real number.
• We must identify which option among the given choices satisfies both inequalities.

Concept / Approach:
We solve each inequality separately to obtain intervals for x. The solution to the compound inequality is the intersection (overlap) of these intervals. Once we know the combined range of x that satisfies both conditions, we test the given options to see which one lies within that range. Working systematically avoids errors and makes the reasoning clear.

Step-by-Step Solution:
Step 1: Solve the first inequality 5x − 4 ≤ 2 − x.Step 2: Add x to both sides: 6x − 4 ≤ 2.Step 3: Add 4 to both sides: 6x ≤ 6.Step 4: Divide by 6 to obtain x ≤ 1.Step 5: Now solve the second inequality 4x + 5 > 2x − 5.Step 6: Subtract 2x from both sides: 2x + 5 > −5.Step 7: Subtract 5 from both sides: 2x > −10.Step 8: Divide by 2: x > −5.Step 9: The combined solution is −5 < x ≤ 1.

Verification / Alternative check:
Now check each option. For x = 3, the first inequality fails because 5(3) − 4 = 11 is not less than or equal to 2 − 3 = −1. For x = 6 and x = −7, the inequalities also do not satisfy both conditions. For x = 0, both inequalities hold, but we must also test the options given in the problem. Among the listed options, x = −1 clearly lies between −5 and 1. Substituting x = −1 in the first inequality gives 5(−1) − 4 = −9 and 2 − (−1) = 3, so −9 ≤ 3 is true. In the second inequality, 4(−1) + 5 = 1 and 2(−1) − 5 = −7, so 1 > −7 is also true. Thus x = −1 satisfies both inequalities.

Why Other Options Are Wrong:
The value 3 is greater than 1 and therefore fails the first inequality. The value 6 is also outside the combined range. The value −7 is less than −5 and fails the second inequality because it does not satisfy x > −5. The value 0 does satisfy both inequalities, but if it is not among the provided options, it cannot be chosen. Among the listed choices, only −1 lies within the interval −5 < x ≤ 1 and satisfies both inequalities.

Common Pitfalls:
One common mistake is to solve only one inequality and forget to intersect the solution sets. Another error is reversing the inequality sign incorrectly when multiplying or dividing by negative numbers. Writing down each step and sketching the intervals on a number line can help you visualise the overlap and avoid mistakes when combining the solutions.

Final Answer:
The value of x that satisfies both inequalities is −1.