Exercise algebra.equations.rational

2*x^2 == 10

x^2 == 5

x == sqrt 5 or x == -sqrt 5

2.

(7*x^2-21)/(2*x^2-5) == 0

⇒ algebra.equations.rational.division-zero, clipboard=[condition == logic1.and (x /= -1/2*sqrt 10) (x /= 1/2*sqrt 10)]

7*x^2-21 == 0

7*x^2 == 21

x^2 == 3

⇒ algebra.equations.rational.division-zero

x == sqrt 3 or x == -sqrt 3

3.

(3*x^2-6)/(4*x^2+1) == 0

3*x^2-6 == 0

3*x^2 == 6

x^2 == 2

x == sqrt 2 or x == -sqrt 2

4.

(4*x^2-24)/(6*x^2-2) == 0

⇒ algebra.equations.rational.division-zero, clipboard=[condition == logic1.and (x /= -1/3*sqrt 3) (x /= 1/3*sqrt 3)]

4*x^2-24 == 0

4*x^2 == 24

x^2 == 6

x == sqrt 6 or x == -sqrt 6

5.

x^2/(x+4) == (3*x+4)/(x+4)

⇒ algebra.equations.rational.same-divisor, clipboard=[condition == (x /= -4)]

x^2 == 3*x+4

x^2-3*x-4 == 0

(x+1)*(x-4) == 0

x == -1 or x == 4

6.

(x^2+2)/(x-2) == (x+8)/(x-2)

⇒ algebra.equations.rational.same-divisor, clipboard=[condition == (x /= 2)]

x^2+2 == x+8

x^2-x-6 == 0

(x+2)*(x-3) == 0

x == -2 or x == 3

7.

(x^2+6*x-6)/(x^2-1) == (4*x+9)/(x^2-1)

⇒ algebra.equations.rational.same-divisor, clipboard=[condition == logic1.and (x /= -1) (x /= 1)]

x^2+6*x-6 == 4*x+9

x^2+2*x-15 == 0

(x-3)*(x+5) == 0

x == 3 or x == -5

8.

(x^2+6)/(x^2-2) == 7*x/(x^2-2)

⇒ algebra.equations.rational.same-divisor, clipboard=[condition == logic1.and (x /= -sqrt 2) (x /= sqrt 2)]

x^2+6 == 7*x

x^2-7*x+6 == 0

(x-1)*(x-6) == 0

x == 1 or x == 6

9.

(x^2+6*x)/(x^2-1) == (3*x+4)/(x^2-1)

⇒ algebra.equations.rational.same-divisor, clipboard=[condition == logic1.and (x /= -1) (x /= 1)]

x^2+6*x == 3*x+4

x^2+3*x-4 == 0

(x-1)*(x+4) == 0

x == 1 or x == -4

x == -4

10.

(x^2+6)/(x-3) == 5*x/(x-3)

⇒ algebra.equations.rational.same-divisor, clipboard=[condition == (x /= 3)]

x^2+6 == 5*x

x^2-5*x+6 == 0

(x-2)*(x-3) == 0

x == 2 or x == 3

x == 2

11.

(x^2+4*x)/(x^2-4) == (3*x+6)/(x^2-4)

⇒ algebra.equations.rational.same-divisor, clipboard=[condition == logic1.and (x /= -2) (x /= 2)]

x^2+4*x == 3*x+6

x^2+x-6 == 0

(x-2)*(x+3) == 0

x == 2 or x == -3

x == -3

12.

(x^2+2*x-4)/(x-5) == (4*x+11)/(x-5)

⇒ algebra.equations.rational.same-divisor, clipboard=[condition == (x /= 5)]

x^2+2*x-4 == 4*x+11

x^2-2*x-15 == 0

(x+3)*(x-5) == 0

x == -3 or x == 5

x == -3

13.

(5*x+2)/(2*x-1) == (5*x+2)/(3*x+5)

⇒ algebra.equations.rational.same-dividend, clipboard=[condition == logic1.and (x /= 1/2) (x /= -5/3)]

5*x+2 == 0 or 2*x-1 == 3*x+5

5*x == -2 or 2*x-1 == 3*x+5

x == -2/5 or 2*x-1 == 3*x+5

x == -2/5 or -x-6 == 0

x == -2/5 or -x == 6

⇒ algebra.equations.coverup.negate

x == -2/5 or x == -6

14.

(x^2-9)/(4*x-1) == (x^2-9)/(2*x+7)

⇒ algebra.equations.rational.same-dividend, clipboard=[condition == logic1.and (x /= 1/4) (x /= -7/2)]

x^2-9 == 0 or 4*x-1 == 2*x+7

x^2 == 9 or 4*x-1 == 2*x+7

x == 3 or x == -3 or 4*x-1 == 2*x+7

x == 3 or x == -3 or 2*x-8 == 0

x == 3 or x == -3 or 2*x == 8

x == 3 or x == -3 or x == 4

15.

(3*x-2)/(2*x^2) == (3*x-2)/(x^2+4)

⇒ algebra.equations.rational.same-dividend, clipboard=[condition == (x /= 0)]

3*x-2 == 0 or 2*x^2 == x^2+4

3*x == 2 or 2*x^2 == x^2+4

x == 2/3 or 2*x^2 == x^2+4

x == 2/3 or x^2-4 == 0

x == 2/3 or x^2 == 4

x == 2/3 or x == 2 or x == -2

16.

(2*x+1)/(x^2+3*x) == (2*x+1)/(5*x+8)

⇒ algebra.equations.rational.same-dividend, clipboard=[condition == logic1.and (logic1.and (x /= -3) (x /= 0)) (x /= -8/5)]

2*x+1 == 0 or x^2+3*x == 5*x+8

2*x == -1 or x^2+3*x == 5*x+8

x == -1/2 or x^2+3*x == 5*x+8

x == -1/2 or x^2-2*x-8 == 0

x == -1/2 or (x+2)*(x-4) == 0

x == -1/2 or x == -2 or x == 4

17.

(x^2-1)/(2*x+2) == (x^2-1)/(x+8)

⇒ algebra.equations.rational.same-dividend, clipboard=[condition == logic1.and (x /= -1) (x /= -8)]

x^2-1 == 0 or 2*x+2 == x+8

x^2 == 1 or 2*x+2 == x+8

x == 1 or x == -1 or 2*x+2 == x+8

x == 1 or x == -1 or x-6 == 0

x == 1 or x == -1 or x == 6

x == 1 or x == 6

18.

(x^2-4)/(3*x-6) == (x^2-4)/(2*x+1)

⇒ algebra.equations.rational.same-dividend, clipboard=[condition == logic1.and (x /= 2) (x /= -1/2)]

x^2-4 == 0 or 3*x-6 == 2*x+1

x^2 == 4 or 3*x-6 == 2*x+1

x == 2 or x == -2 or 3*x-6 == 2*x+1

x == 2 or x == -2 or x-7 == 0

x == 2 or x == -2 or x == 7

⇒ algebra.equations.polynomial.power-factors

x == -2 or x == 7

19.

(x^2+5*x)/(2*x^2) == (x^2+5*x)/(x^2+4)

⇒ algebra.equations.rational.same-dividend, clipboard=[condition == (x /= 0)]

x^2+5*x == 0 or 2*x^2 == x^2+4

x == 0 or x+5 == 0 or 2*x^2 == x^2+4

x == 0 or x == -5 or 2*x^2 == x^2+4

x == 0 or x == -5 or x^2-4 == 0

x == 0 or x == -5 or x^2 == 4

x == 0 or x == -5 or x == 2 or x == -2

⇒ algebra.equations.polynomial.power-factors

x == -5 or x == 2 or x == -2

20.

(x^2-3*x)/(2*x-6) == (x^2-3*x)/(4*x+2)

⇒ algebra.equations.rational.same-dividend, clipboard=[condition == logic1.and (x /= 3) (x /= -1/2)]

x^2-3*x == 0 or 2*x-6 == 4*x+2

x == 0 or x-3 == 0 or 2*x-6 == 4*x+2

⇒ algebra.equations.quadratic.same-con-factor

x == 0 or x == 3 or 2*x-6 == 4*x+2

x == 0 or x == 3 or x-3 == 2*x+1

x == 0 or x == 3 or -x-4 == 0

x == 0 or x == 3 or -x == 4

⇒ algebra.equations.coverup.negate

x == 0 or x == 3 or x == -4

x == 0 or x == -4

21.

x/(x+1) == 1+3/4

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == (x /= -1)]

4*x == (x+1)*7

-(x+1)*7+4*x == 0

-7*x-7+4*x == 0

-3*x-7 == 0

-3*x == 7

⇒ algebra.equations.quadratic.same-con-factor

x == -7/3

22.

(x+2)/(3*x) == 1+1/3

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == (x /= 0)]

(x+2)*3 == 12*x

x+2 == 4*x

-3*x+2 == 0

-3*x == -2

x == 2/3

23.

(2*x+3)/(x-1) == 3+1/2

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == (x /= 1)]

(2*x+3)*2 == (x-1)*7

(2*x+3)*2-(x-1)*7 == 0

4*x+6-7*x+7 == 0

-3*x+13 == 0

-3*x == -13

x == 13/3

24.

(x-3)/(1-x) == 1+2/5

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == (x /= 1)]

(x-3)*5 == (1-x)*7

(x-3)*5-(1-x)*7 == 0

5*x-22+7*x == 0

12*x-22 == 0

12*x == 22

x == 11/6

25.

(x+4)/(x+3) == (x+1)/(x+2)

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == logic1.and (x /= -3) (x /= -2)]

(x+4)*(x+2) == (x+3)*(x+1)

(x+4)*(x+2)-(x+3)*(x+1) == 0

x^2+6*x+8-x^2-4*x-3 == 0

2*x+5 == 0

2*x == -5

x == -5/2

26.

(2*x+3)/(x-1) == (2*x-1)/(x-2)

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == logic1.and (x /= 1) (x /= 2)]

(2*x+3)*(x-2) == (x-1)*(2*x-1)

(2*x+3)*(x-2)-(x-1)*(2*x-1) == 0

2*x^2-x-6-2*x^2+3*x-1 == 0

2*x-7 == 0

2*x == 7

x == 7/2

27.

(3*x+6)/(3*x-1) == (x+4)/(x+1)

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == logic1.and (x /= 1/3) (x /= -1)]

(3*x+6)*(x+1) == (3*x-1)*(x+4)

(3*x+6)*(x+1)-(3*x-1)*(x+4) == 0

3*x^2+9*x+6-3*x^2-11*x+4 == 0

-2*x+10 == 0

-2*x == -10

x == 5

28.

(x+2)/(2*x+5) == (x+4)/(2*x-3)

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == logic1.and (x /= -5/2) (x /= 3/2)]

(x+2)*(2*x-3) == (2*x+5)*(x+4)

(x+2)*(2*x-3)-(2*x+5)*(x+4) == 0

2*x^2+x-6-2*x^2-13*x-20 == 0

-12*x-26 == 0

-12*x == 26

x == -13/6

29.

(x+5)/(2*x)+2 == 5

⇒ algebra.equations.coverup.plus

(x+5)/(2*x) == 3

⇒ algebra.equations.rational.multiply-one-div, clipboard=[condition == (x /= 0)]

x+5 == 6*x

-5*x+5 == 0

-5*x == -5

⇒ algebra.equations.coverup.minus-left

x == 1

30.

(3*x+4)/(x+2)-3 == 2

(3*x+4)/(x+2) == 5

⇒ algebra.equations.rational.multiply-one-div, clipboard=[condition == (x /= -2)]

3*x+4 == (x+2)*5

-(x+2)*5+3*x+4 == 0

-5*x-10+3*x+4 == 0

-2*x-6 == 0

-2*x == 6

x == -3

31.

x^2/(5*x+6)+4 == 5

⇒ algebra.equations.coverup.plus

x^2/(5*x+6) == 1

⇒ algebra.equations.rational.division-one, clipboard=[condition == (x /= -6/5)]

x^2 == 5*x+6

x^2-5*x-6 == 0

(x+1)*(x-6) == 0

x == -1 or x == 6

32.

x^2/(2*x-3)+3 == 7

⇒ algebra.equations.coverup.plus

x^2/(2*x-3) == 4

⇒ algebra.equations.rational.multiply-one-div, clipboard=[condition == (x /= 3/2)]

x^2 == (2*x-3)*4

-(2*x-3)*4+x^2 == 0

-8*x+12+x^2 == 0

(x-2)*(x-6) == 0

x == 2 or x == 6

33.

(x-2)/(x-3) == x/2

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == (x /= 3)]

(x-2)*2 == (x-3)*x

(x-2)*2-(x-3)*x == 0

2*x-4-x^2+3*x == 0

5*x-4-x^2 == 0

x^2-5*x+4 == 0

(x-1)*(x-4) == 0

x == 1 or x == 4

34.

(x+9)/(x-5) == 2/x

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == logic1.and (x /= 5) (x /= 0)]

(x+9)*x == (x-5)*2

(x+9)*x-(x-5)*2 == 0

x^2+9*x-2*x+10 == 0

x^2+7*x+10 == 0

(x+2)*(x+5) == 0

x == -2 or x == -5

35.

(x+2)/(x+4) == 2/(x+1)

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == logic1.and (x /= -4) (x /= -1)]

(x+2)*(x+1) == (x+4)*2

(x+2)*(x+1)-(x+4)*2 == 0

x^2+3*x+2-2*x-8 == 0

x^2+x-6 == 0

(x-2)*(x+3) == 0

x == 2 or x == -3

36.

(-3)/(x-5) == (x+3)/(x+1)

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == logic1.and (x /= 5) (x /= -1)]

-3*(x+1) == (x-5)*(x+3)

-3*(x+1)-(x-5)*(x+3) == 0

-3*x-3-x^2+2*x+15 == 0

-x+12-x^2 == 0

x^2+x-12 == 0

(x-3)*(x+4) == 0

x == 3 or x == -4

37.

(x+1)/(x+2) == (7*x+1)/(2*x-4)

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == logic1.and (x /= -2) (x /= 2)]

(x+1)*(2*x-4) == (x+2)*(7*x+1)

(x+1)*(2*x-4)-(x+2)*(7*x+1) == 0

2*x^2-2*x-4-7*x^2-15*x-2 == 0

-5*x^2-17*x-6 == 0

5*x^2+17*x+6 == 0

⇒ algebra.equations.quadratic.abc, clipboard=[D == 169, a == 5, b == 17, c == 6, condition == logic1.and (x /= -2) (x /= 2)]

x == -2/5 or x == -3

38.

(2*x-7)/(5-x) == (x+1)/(3*x-7)

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == logic1.and (x /= 5) (x /= 7/3)]

(2*x-7)*(3*x-7) == (5-x)*(x+1)

(2*x-7)*(3*x-7)-(5-x)*(x+1) == 0

6*x^2-35*x+49-4*x-5+x^2 == 0

7*x^2-39*x+44 == 0

⇒ algebra.equations.quadratic.abc, clipboard=[D == 289, a == 7, b == -39, c == 44, condition == logic1.and (x /= 5) (x /= 7/3)]

x == 4 or x == 11/7

39.

(x+1)/(x-1) == (3*x-7)/(x-2)

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == logic1.and (x /= 1) (x /= 2)]

(x+1)*(x-2) == (x-1)*(3*x-7)

(x+1)*(x-2)-(x-1)*(3*x-7) == 0

x^2-x-2-3*x^2+10*x-7 == 0

-2*x^2+9*x-9 == 0

2*x^2-9*x+9 == 0

⇒ algebra.equations.quadratic.abc, clipboard=[D == 9, a == 2, b == -9, c == 9, condition == logic1.and (x /= 1) (x /= 2)]

x == 3 or x == 3/2

40.

(3*x-7)/(x-2) == (7-x)/(3*x-3)

⇒ algebra.equations.rational.cross-multiply, clipboard=[condition == logic1.and (x /= 2) (x /= 1)]

(3*x-7)*(3*x-3) == (x-2)*(7-x)

(3*x-7)*(3*x-3)-(x-2)*(7-x) == 0

9*x^2-30*x+21-9*x+x^2+14 == 0