2011 Λύσεις Σχ. β&I

X 1 3 x 3 1 3x 1 x 3 3x x 1 3 2x 2 x 1 x 1 2 x 2 1 2x

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x – 1 3 x 3 – 1 3x – 1 ≤ x – 3 3x – x ≤ 1 – 3 2x ≤ –2 x ≤ –1. x – 1 2 > x 2 + 1 2x – 1 > x + 2 2x – x > 1 + 2 x > 3. x – 2 2 + 1 – 2x 5 < x 10 2 5 5x – 10 + 2 – 4x < x – 4 x – 12 2 + x 2 + 3 4 > x 2(x – 12) + 2x + 3 > 4x x – 1 2 + 2x + 3 4 < x 6 6(x – 1) + 3(2x + 3) < 2x
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4. ñ ñ √È ·ÓÈÛÒÛÂȘ Û˘Ó·ÏËıÂ‡Ô˘Ó ÁÈ· x (– 1 7 , 7 3 ). √È ·Î¤Ú·È˜ ÙÈ̤˜ ÙÔ˘ x ÛÙÔ ‰È¿ÛÙËÌ· ·˘Ùfi Â›Ó·È ÔÈ 0, 1, 2. 5. i) |x| < 3 –3 < x < 3. ÕÚ· x (–3, 3). ii) |x – 1| ≤ 4 –4 ≤ x – 1 ≤ 4 1 – 4 ≤ x ≤ 1 + 4 –3 ≤ x ≤ 5. ÕÚ· x [–3, 5]. iii) |2x + 1| < 5 –5 < 2x + 1< 5 – 5 –1 < 2x < 5 – 1 –6 < 2x < 4 –3 < x < 2. ÕÚ· x (–3, 2). 6. i) |x| ≥ 3 x ≤ –3 x ≥ 3. ÕÚ· x (– , –3] [3, + ). ii) |x – 1| > 4 x – 1 < –4 x – 1 > 4 x < –3 x > 5. ÕÚ· x (– , –3) (5, + ). iii) |2x + 1| ≥ 5 2x + 1 ≤ –5 2x + 1 ≥ 5 2x ≤ –6 2x ≥ 4 x ≤ –3 x ≥ 2. ÕÚ· x (– , –3] [2, + ). 7. i) ∞fi ÙÔÓ ÔÚÈÛÌfi Ù˘ ·fiÏ˘Ù˘ ÙÈÌ‹˜ ¤¯Ô˘Ì |·| = · · ≥ 0. ∂Ô̤ӈ˜ |2x – 6| = 2x – 6 2x – 6 ≥ 0 2x ≥ 6 x ≥ 3. ii) |3x – 1| = 1 – 3x 3x – 1 ≤ 0 3x ≤ 1 x ≤ 1 3 . 8. i) 3|x – 1| – 12 + 10 < 2|x – 1| |x – 1| < 2 –2 < x – 1 < 2 –1 < x< 3. ÕÚ· x (–1, 3). |x – 1| – 4 2 + 5 3 < |x – 1| 3 3 |x – 1| – 4 + 10 < 2|x – 1| 2x + x < 8 – 1 3x < 7 x < 7 3 . x – 4 + x + 1 2 < 0 2x – 8 + x + 1 < 0 7x > –1 x > – 1 7 . 2x – x – 1 8 > x 16x – x + 1 > 8x 16x – x – 8x > –1 ∫∂º∞§∞π√ 4: ∞¡π™ø™∂π™ 48
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ii) 3|x| + 3 – 4|x| > 2 – 2|x| 3|x| – 4|x| + 2|x| > 2 – 3 |x| > –1 Ô˘ ·ÏËı‡ÂÈ ÁÈ· οı x . 9. |x – 3| ≤ 5 –5 ≤ x – 3 ≤ 5 3 – 5 ≤ x ≤ 5 + 3 –2 ≤ x ≤ 8. ÕÚ· x [–2, 8]. 10. ΔÔ Î¤ÓÙÚÔ ÙÔ˘ ‰È·ÛÙ‹Ì·ÙÔ˜ (– 7, 3) Â›Ó·È ÙÔ –7 + 3 2 = –2. Œ¯Ô˘Ì x (–7, 3) –7 < x < 3 –7 – (–2) < x – (–2) < 3 – (–2) –7 + 2 < x + 2 < 3 + 2 –5 < x + 2 < 5 | x + 2| < 5. 11. 41 ≤ 9 5 C + 32 ≤ 50 41 – 32 ≤ 9 5 C ≤ 50 – 32 9 ≤ 9 5 C ≤ 18 5 ≤ C ≤ 10. μã √ª∞¢∞™ 1. i) 3 ≤ 4x – 1 ≤ 6 3 ≤ 4x – 1 Î·È 4x – 1 ≤ 6. ∑ËÙ¿Ì ÂÔ̤ӈ˜ ÙȘ ÙÈ̤˜ ÙÔ˘ x ÁÈ· ÙȘ Ôԛ˜ Û˘Ó·ÏËıÂ‡Ô˘Ó ÔÈ ·ÓÈÛÒÛÂȘ 3 ≤ 4x – 1 Î·È 4x – 1 ≤ 6. ñ 3 ≤ 4x – 1 4 ≤ 4x 4x ≥ 4 x ≥ 1. ñ 4x – 1 ≤ 6 4x ≤ 7 x ≤ 7 4 . ÕÚ· x [1, 7 4 ]. ii) –4 ≤ 2 – 3x ≤ –2 –4 ≤ 2 – 3x Î·È 2 –3x ≤ –2.
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