1.Algebra Booster

Quadratic Equations and Expressions 2.47
fi
Ï x ≥ 1
Ô
Ì x > 1
Ô 2
Óx
- 4x
+ 3>
0
fi
Ï x ≥ 1
Ô
Ì x > 1
Ô
Ó( x -1)( x - 3) > 0
fi
Ï x ≥ 1
Ô
Ì x > 1
Ô
Óx< 1, x>
3
Hence the solution set is (3, ).
240. We have,
3 5x
- 4 > x
fi 5x – 4 > x 3
fi 5x – 4 – x 3 > 0
fi x 3 – 5x + 4 > 0
fi x 3 – x 2 + x 2 – x – 4x + 4 > 0
fi x 2 (x – 1) + x(x – 1) – 4(x – 1) > 0
fi (x – 1)(x 2 + x – 4) > 0
1 17 1 17
fi
- - < x< 1, x>
- +
2 2
Hence the solution set is
Ê-1- 17 ˆ Ê- 1+
17 ˆ
Á ,1 ,
Ë
˜ Á ˜
2 ¯ Ë 2 ¯
241. We have,
3 2 1/3
3 x - 7 x > ( - 36)
fi x 3 – 7x 2 > –36
fi x 3 – 7x 2 + 36 > 0
fi x 3 + 2x 2 – 9x 2 – 18x + 18x + 36 > 0
fi x 2 (x + 2) – 9x(x + 2) + 18(x + 2) > 0
fi (x + 2)(x 2 – 9x + 18) > 0
fi (x + 2)(x 2 – 9x + 18) > 0
fi (x + 2)(x – 3)(x – 6) > 0
fi –2 < x < 3, x > 6
Hence the solution set is (–2, 3) » (6, ).
242. We have,
2
- x + 4x - 3 > (6-2 x)
fi
Ï
2 2 2
Ô- x + 4x - 3 > (6-2 x) ÏÔ- x + 4x
-3≥0
Ì
and Ì
ÔÓ
6-2x≥0 ÔÓ
6- 2x<
0
fi
Ï
2 2
Ô5x - 28x + 39< 0 ÏÔx - 4x
+ 3£
0
Ì
and Ì
ÔÓ
6- 2x≥ 0 ÔÓ
3- x<
0
fi
Ï
2
Ô5x -15x - 13x+ 39<
0 Ï( x -1)( x - 3) £ 0
Ì
and Ì
ÔÓ
6-2x
≥0
Ó x –3>
0
fi
Ï( x -3)(5x - 13) < 0 Ï1 £ x£
3
Ì
and Ì
Ó 6- 2x≥ 0 Ó x>
3
Ï13
Ô < x < 3
Ï1£ x £ 3
fi Ì 5
and Ì
Ô x > 3
x £ 3
Ó
Ó
Ê13 ˆ
Hence the solution set is Á ,3 .
Ë
˜
5 ¯
243. We have,
x - 2 + x - 1>
2
fi
Ï
x –2≥
0
Ô
Ì
x –1≥
0
Ô
Óx - 2 + x - 1 + 2 x - 2 x - 1 > 4
fi
Ï x ≥ 2
Ô
Ì x ≥ 1
Ô
Ó2 x - 2 x - 1 > (7–2 x)
fi
Ï x ≥ 2
Ô
Ì x ≥ 1
Ô 2 2
Ó4( x - 3x + 2) > (7–2 x)
fi
Ï
x ≥ 2
Ô
Ì
x ≥ 1
Ô 2 2
Ó4( x - 3x + 2) > 4x - 28x
+ 49
fi
Ï x ≥ 2
Ô
Ì x ≥ 1
Ô
Ó16x
> 41
Ï x ≥ 2
Ô
fi Ì x ≥ 1
Ô
Óx
> 41/16
Ê41 ˆ
Hence, the solution set is Á , .
Ë
˜
16 ¯
244. We have,
x -6 - 10- x ≥1
fi
fi
fi
fi
fi
fi
Ï x - 6≥
0
Ô
Ì 10 - x £ 0
Ô
Ó x - 6 ≥ 1+ 10-
x
Ï x ≥ 6
Ô
Ì 10 £ x
Ô
Óx - 6 ≥ 1 + 10 - x + 2 10 - x
Ï x ≥ 6
Ô
Ì 10 £ x
Ô
Ó2x
-17≥ 2 10-
x
Ï x ≥ 6
Ô
Ì 10 £ x
Ô 2
Ó(2x
-17) ≥ 4(10 – x)
Ï x ≥ 6
Ô
Ì 10 £ x
Ô 2
Ó4x - 68x + 289 ≥ 40 - 4x
Ï x ≥ 6
Ô
Ì 10 £ x
Ô 2
Ó4x
- 64x
+ 249 ≥ 0