3.4 Linear Inequalities - 章节练习

答案：
\(2(x - 3) \geq 0\)
\(x - 3 \geq 0\)
\(x \geq 3\)

答案：
\(8(1 - x) > x - 1\)
\(8 - 8x > x - 1\)
\(8 + 1 > x + 8x\)
\(9 > 9x\)
\(1 > x\) 或 \(x < 1\)

答案：
\(3(x + 7) \leq 8 - x\)
\(3x + 21 \leq 8 - x\)
\(3x + x \leq 8 - 21\)
\(4x \leq -13\)
\(x \leq -\frac{13}{4} = -3.25\)

答案：
\(2(x - 3) - (x + 12) < 0\)
\(2x - 6 - x - 12 < 0\)
\(x - 18 < 0\)
\(x < 18\)

答案：
第一个不等式：\(3(x - 2) > x - 4\)
\(3x - 6 > x - 4\)
\(2x > 2\)
\(x > 1\)

第二个不等式：\(4x + 12 > 2x + 17\)
\(2x > 5\)
\(x > 2.5\)

交集：\(x > 2.5\)

答案：
第一个不等式：\(2x - 5 < x - 1\)
\(x < 4\)

第二个不等式：\(7(x + 1) > 23 - x\)
\(7x + 7 > 23 - x\)
\(8x > 16\)
\(x > 2\)

交集：\(2 < x < 4\)

答案：
第一个不等式：\(5x + 3 < 9\)
\(5x < 6\)
\(x < 1.2\)

第二个不等式：\(5(2x + 1) > 27\)
\(10x + 5 > 27\)
\(10x > 22\)
\(x > 2.2\)

并集：\(x < 1.2\) 或 \(x > 2.2\)

答案：
首先求各集合：
\(A = \{ x: 3x + 5 > 2 \} = \{ x: 3x > -3 \} = \{ x: x > -1 \}\)
\(B = \{ x: \frac{x}{2} + 1 \leq 3 \} = \{ x: \frac{x}{2} \leq 2 \} = \{ x: x \leq 4 \}\)
\(C = \{ x: 11 < 2x - 1 \} = \{ x: 2x > 12 \} = \{ x: x > 6 \}\)

\(B \cup C = \{ x: x \leq 4 \} \cup \{ x: x > 6 \} = \{ x: x \leq 4 \ or \ x > 6 \}\)
\(A \cap (B \cup C) = \{ x: x > -1 \} \cap \{ x: x \leq 4 \ or \ x > 6 \} = \{ x: -1 < x \leq 4 \} \cup \{ x: x > 6 \}\)

因此：\(p = -1\), \(q = 4\), \(r = 6\)