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If 4x - 7 < x - 2 and 5x + 2/3 ≥ 3x + 1, then which of the following values of x satisfies both inequalities?

Difficulty: Medium

Correct Answer: 1

Explanation:


Introduction / Context:
This question tests solving a pair of linear inequalities and then checking which option satisfies both conditions at the same time. You must treat each inequality separately, find the range of x allowed by each, and finally take the common intersection of those ranges. Only values from the options that lie in this intersection are valid answers.


Given Data / Assumptions:

  • Inequality 1: 4x - 7 < x - 2.
  • Inequality 2: 5x + 2/3 ≥ 3x + 1.
  • We must pick x from the given options that satisfies both inequalities.


Concept / Approach:
The method is to solve each inequality by standard algebraic manipulation, moving variable terms to one side and constants to the other. This produces intervals for x. The intersection of these intervals gives the set of values that satisfy both conditions. Then we see which option lies in that set.


Step-by-Step Solution:

Step 1: Solve the first inequality 4x - 7 < x - 2. Step 2: Subtract x from both sides: 3x - 7 < -2. Step 3: Add 7 to both sides: 3x < 5. Step 4: Divide by 3: x < 5/3. Step 5: Solve the second inequality 5x + 2/3 ≥ 3x + 1. Step 6: Subtract 3x from both sides: 2x + 2/3 ≥ 1. Step 7: Subtract 2/3 from both sides: 2x ≥ 1 - 2/3 = 1/3. Step 8: Divide by 2: x ≥ 1/6. Step 9: Combine the results from both inequalities: x must satisfy x ≥ 1/6 and x < 5/3. Step 10: So the solution interval is 1/6 ≤ x < 5/3. Step 11: Check each option: 2 is greater than 5/3, so it does not satisfy the first inequality. -1 and -2 are less than 1/6, so they do not satisfy the second inequality. Only x = 1 lies between 1/6 and 5/3.


Verification / Alternative check:
Substitute x = 1 directly into both inequalities. For the first: 4(1) - 7 = -3 and 1 - 2 = -1, so -3 < -1 is true. For the second: 5(1) + 2/3 = 5 + 2/3 and 3(1) + 1 = 4, so 5+2/3 ≥ 4 is also true. Therefore, x = 1 satisfies both inequalities simultaneously.


Why Other Options Are Wrong:

2: fails the first inequality because 4*2 - 7 = 1 is not less than 2 - 2 = 0. -1: fails the second inequality because 5(-1) + 2/3 is negative and cannot be greater than or equal to 3(-1) + 1 which is also negative but larger. -2: similarly fails the second inequality, giving a much smaller left side than right side. 0: is greater than 1/6? No, it is not, so it does not satisfy the second inequality either.


Common Pitfalls:
One common mistake is dropping the fraction 2/3 incorrectly or miscalculating 1 - 2/3. Another pitfall is misunderstanding the combination of inequalities and treating them independently without intersecting the solution sets. Carefully solving and then checking against the interval helps avoid such issues.


Final Answer:
1

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