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Question

# Solve : (i) 13x−6=52(ii) 2x3−3x8=712(iii) (x+2)(x+3)+(x−3)(x−2)−2x(x+1)=0(iv) 110−7x=35(v) 13(x−4)−3(x−9)−4(x+4)=0(vi) x+7−8x3=17x6−5x8(vii) 3x−24−2x+33=23−x(viii) x+26−(11−x3−14)=3x−412(ix) 25x−53x=115(x) x+23−x+15=x−34−1(xi) 3x−23+2x+32=x+76(xii) x−x−12=1−x−23(xiii) 9x+72−(x−x−27)=36(xiv) 6x+12+1=7x−33

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Solution

## (i) 13x−6=52⇒ 13x=52+61⇒ 13x=5×12×1+6×21×2⇒ 13x=52+122⇒ 13x=5+122=13x=172.=x=17×32×1=512=2512(ii) 2x3−3x8=71223,823,423,233,11 L.C.M. of 3 and 8 =2×2×2×3=24∴ 2x×83×8−3x×88×3=712=16x24−9x24=712=16x−9x24=712=7x24=712=x=7×2712×7=2∴ x=2(iii) (x+2)(x+3)+(x−3)(x−2)−2x(x+1)=0⇒ [x2+(2+3)x+2×3]+[x2+(−3−2)x+(−3)(−2)]−2x2−2x=0⇒ x2+5x+6+x2−5x+6−2x2−2x=0⇒ x2+x2−2x2+5x−5x−2x+6+6=0=−2x+12=0Subtracting 12 from both sides,−2x+12−12=0−12⇒ −2x=−12Dividing by -2−2x−2=−12−2⇒ x=6∴ x=6VerificationL.H.S.=(x+2)(x+3)+(x−3)(x−2)−2x(x+1)=(6+2)(6+3)+(6−3)(6−2)−2×6(6+1)=8×9+3×4−12×7=72+12−84=84−84=0=R.H.S.(iv) 110−7x=35 ⇒ −7x=35−110 ⇒ −7x=35×101×10−1×110×1 ⇒ −7x=350−110 ⇒ 1x=350−110×(−7) ⇒ x=349(−70)=−70349(v) 13(x−4)−3(x−9)−4(x+4)=0⇒ 13(x−4)−3(x−9)−5(x+4)=0⇒ 13x−52−3x+27−5x−20=0⇒ 13x−3x−5x−52+27−20=0⇒ 13x−8x−72+27=0⇒ 5x−45=0Dividing by 5,5x5−455=0⇒ x−9=0⇒ x=9Verification,~L.H.S. =13(x−4)−3(x−9)−5(x+4)=13(9−4)−3(9−9)−5(9+4)=13×5−3×0−5×13(vi) x+7−8x3=17x6−5x8⇒ 3(x+7)−8x3=17x×4−5x×324⇒ 3x+21−8x3=68x−15x24⇒ −5x+213=53x24⇒ 3×53x=24(−5x+21)⇒ 159x=−120x+504⇒ 159x+120x=504⇒ 279x=504⇒ x=504279=16893=5661∴ x=12531(vii) 3x−24−2x+33=23−x=3x−24−2x+33=23−x1=3(3x−2)−4(2x+3)12=2×13×1−x×31×3=9x−6−8x−1212=2−3x3=(x−18)12=2−3x3=3(x−18)=12(2−3x)=3x−54=24−36x=3x+36x=24+54=39x=78x=7839=2∴ x=2(viii) x+26−(11−x3−14)=3x−412⇒ x+26−(4(11−x)−1×312)=3x−412⇒ x+26−44+4x+312=3x−412⇒ 2(x+2)−41+4x12=3x−412⇒ 2x+4−41+4x12=3x−412⇒ 6x−3712=3x−412⇒ 12(6x−37)=12(3x−4)⇒ 72x−444=36x−48⇒ 72x−36x=−48+444⇒ 36x=396⇒ x=39636=11∴ x=11(ix) 25x−53x=11525x−53x=115⇒ 2×35x×3−5×53x×5=115⇒ 6−2515x=115⇒ −1915x=115⇒ −19x=1515⇒ −19=x∴ x=−19(x) x+23−x+15=x−34−1 (L.C.M. of 3 and 5 = 15)⇒ 5(x+2)−3(x+1)15=x−3−44⇒ 5x+10−3x−315=x−74⇒ 2x+715=x−74⇒ 4(2x+7)=15(x−7)⇒ 8x+28=15x−105⇒ 8x−15x=−133x=−133−7∴ x=19(xi) 3x−23+2x+32=x+76⇒ 2(3x−2)+3(2x+3)6=x+76⇒ 6x−4+6x+96=6x+76⇒ 12x+56=6x+76⇒ 6(12x+5)=6(6x+7)⇒ 72x+30=36x−42⇒ 72x−36x=42−30⇒ 36x=12x=1236∴ x=13(xii) x−x−12=1−x−23⇒ 2(x)−1(x−1)2=3(1)−1(x−2)3⇒ 2x−x+12=3−x+23⇒ 1x+12=5−x3⇒ 3(x+1)=2(5−x)⇒ 3x+3=10−2x⇒ 3x+2x=10−3⇒ 5x=7∴ x=75(xiii) 9x+72−(x−x−27)=36⇒ 9x+72−(7×x−1(x−2)7)=36⇒ 9x+72−(7x−x−27)=36⇒ 9x+72−(6x−27)=36⇒ 7(9x+7)+2(−6x+2)14=36⇒ 63x+49−12x+414=36⇒ 51x+5314=36⇒ 51x+53=14×36⇒ 51x=504−53⇒ 51x=459⇒ x=45951∴ x=9(xiv) 6x+12+1=7x−33⇒ (6x+1)2+1=7x−33⇒ 6x+1+22=7x−33⇒ 6x+32=7x−33⇒ 3(6x+3)=2(7x−3)⇒ 18x+9=14x−6⇒ 18x−14x=−6−9⇒ 4x=−15∴ x=−154

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