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Consider the following system of inequalities. $$x_1 − x_2 ≤ 3\\ x_2 − x_3 ≤ −2,\\ x_3 − x_4 ≤ 10,\\ x_4 − x_2 ≤ α,\\ x_4 − x_3 ≤ −4,$$ where $\alpha$ is a real number. A value for $\alpha$ for which the system has a solution is (a) $-16$, (b)$-12$, (c)$-10$, (d) None.

I have no idea how to approach this problem.

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(a) is true $\implies$ (b) is true $\implies$ (c) is true, so all you need is to test whether (c) holds to decide between it and (d). In any case, it is not hard to see that adding the middle three inequalities gives $\alpha \ge -8$.

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