TOPIC -MODULUS BY ATUL KUMAR FACULTY MATHS ( IIT JEE ) MOB:09310556579, ID-atul.mod@gmail.com, skype id-atul.kumar4 ,73 GR FLR,PKT –B5,VIVEKANAND SOCIETY Sector-8, Rohini, New Delhi-110085 MODULUS: (CONCEPTS) 1. <−≥=00xxxxx.0Rxx∈∀≥⇒ Also 2xx= i. e x2=|x|2RxaxRxaxRxaxaxaxaxoraxaxaxaax∉⇒−=∉⇒−<∈⇒−>±=⇒=>−<⇒><<−⇒<. 2. If a>0, then 3. If a,α∈R and a>0 then ααααααααα+−+=⇒=−∞+∪+−−∞⇒>−++−∈⇒<−aoraxaxaaxaxaaxax),(),(),( 4.If 0⇒−=−xyandyxyxyx (It means that x and y has same sign) 0≥>⇒−=−xyandxyxyyx (It means that x and y has same sign) 6.If a<0, |x|=a⇒no solution |x|a⇒ x∈(-∞ ,∞) 7. |x-k| means the distance of x from k on the number line . |x-1|=3(means the distance of x from 1 on the number line is 3)⇒x=4 or-2 -2 -1 0 1 2 3 4 TOPIC -MODULUS BY ATUL KUMAR FACULTY MATHS ( IIT JEE ) MOB:09310556579, ID-atul.mod@gmail.com, skype id-atul.kumar4 ,73 GR FLR,PKT –B5,VIVEKANAND SOCIETY Sector-8, Rohini, New Delhi-110085 Solve the following: 1. |x-3|= 4 2. |x+2|= 6 3. |x-3/4|=2 4. |x+1/2|= 1 5. |x-1/7|= 2/7 6. 5154x=−|| 7. |3x|= 2 8. 32x= 9. |x-1|>1 10. |x-2|<3 11. |x+1|>2 12. |x-3|< 7 13. |x|<2 14. |x|>3 15. |x-4|>1 16. 1<|x-4|<3 17. 1<|x-1|<2 18. 2<|x-2|<5 19. 4<|x-3| 20. |x-0.3|<0.7 21. 32<3734x<− 22. 109101x103≤+< 23. 2x31≤−≤|| 24. 252x321≤< 25. 613x≥+|| ANSWER 1. 7 or-1 2. 4or-8 3. 11/4or-5/4 4. -3/2or1/2 5. 3/7or-1/7 6. x=1 or3/5 7. x= 2/3 or-2/3 8. x=6 or-6 9. ),(),(∞∪−∞∈20x 10.),(51x−∈ 11.),(),(∞∪−−∞∈13x 12),(104x−∈. 13.),(22x−∈ 14.),(),(∞∪−−∞∈33x 15.),(),(∞∪−∞∈53x 16.),(),(7531x∪∈ 17.),(),(3202x∪−∈ 18.),(),(7403x∪−∈ 19.),(),(∞∪−−∞∈71x 20.),.(140x−∈ 21.)/,()/,(3112321x∪−∈ 22.[)(]5451521x/,//,∪−−∈ 23.[]∪∈21x,[]54, 24.[)(]35313135x/,//,/∪−−∈ 25.),[],(∞−∪−−∞∈719x