DAT Quantitative Reasoning Question of the Day

1. $\frac{8}{3}$ (correct answer)
2. $\frac{10}{3}$
3. $\frac{11}{3}$
4. $\frac{13}{3}$
5. $\frac{14}{3}$

Explanation: We need to consider different cases based on the critical points where expressions inside absolute values equal zero: $x = -1$ and $x = \frac{5}{2}$. Case 1: $x < -1$. Here $2x - 5 < 0$ and $x + 1 < 0$, so $|2x - 5| = -(2x - 5) = 5 - 2x$ and $|x + 1| = -(x + 1) = -x - 1$. The equation becomes $(5 - 2x) + (-x - 1) = 8$, which gives $4 - 3x = 8$, so $x = -\frac{4}{3}$. Since $-\frac{4}{3} > -1$, this doesn't satisfy $x < -1$. Case 2: $-1 \leq x < \frac{5}{2}$. Here $2x - 5 < 0$ and $x + 1 \geq 0$, so $|2x - 5| = 5 - 2x$ and $|x + 1| = x + 1$. The equation becomes $(5 - 2x) + (x + 1) = 8$, which gives $6 - x = 8$, so $x = -2$. Since $-2 < -1$, this doesn't work for this case. Case 3: $x \geq \frac{5}{2}$. Here both expressions are positive, so $|2x - 5| = 2x - 5$ and $|x + 1| = x + 1$. The equation becomes $(2x - 5) + (x + 1) = 8$, which gives $3x - 4 = 8$, so $x = 4$. Since $4 > \frac{5}{2}$, this is valid. Let me recalculate Case 1 more carefully: For $x \leq -1$, we have $5 - 2x - x - 1 = 8$, so $4 - 3x = 8$, giving $x = -\frac{4}{3} \approx -1.33$. Since $-\frac{4}{3} < -1$, this is valid. For Case 2 ($-1 < x < \frac{5}{2}$): $5 - 2x + x + 1 = 8$, so $6 - x = 8$, giving $x = -2$. Since $-2 < -1$, this belongs to Case 1, not Case 2. So our solutions are $x = -\frac{4}{3}$ and $x = 4$. Their sum is $-\frac{4}{3} + 4 = -\frac{4}{3} + \frac{12}{3} = \frac{8}{3}$.