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3 x 2 + 4 x + 5 ( x + 6 ) ( x 2 ) + 2 x 2 + x + 6 x 2 + 4 x 12 x 2 4 x 6 x 2 + 4 x 12 Factor the denominators to determine if they're the same . 3 x 2 + 4 x + 5 ( x + 6 ) ( x 2 ) + 2 x 2 + x + 6 ( x + 6 ) ( x 2 ) x 2 4 x 6 ( x + 6 ) ( x 2 ) The denominators are the same . Combine the numerators being careful to note the negative sign . 3 x 2 + 4 x + 5 + 2 x 2 + x + 6 ( x 2 4 x + 6 ) ( x + 6 ) ( x 2 ) 3 x 2 + 4 x + 5 + 2 x 2 + x + 6 x 2 + 4 x + 6 ( x + 6 ) ( x 2 ) 4 x 2 + 9 x + 17 ( x + 6 ) ( x 2 )

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Practice set a

Add or Subtract the following rational expressions.

5 x 2 y 2 3 x 2 y 2

x y 2

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x + y x y + 2 x + 3 y x y

3 x + 4 y x y

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4 x 2 x + 4 3 x + 10 x 2 + 2 x + 5 3 x + 10

3 x 2 3 x 1 3 x + 10

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x ( x + 1 ) x ( 2 x + 3 ) + 3 x 2 x + 7 2 x 2 + 3 x

4 x 2 + 7 x ( 2 x + 3 )

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4 x + 3 x 2 x 6 8 x 4 ( x + 2 ) ( x 3 )

4 x + 7 ( x + 2 ) ( x 3 )

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5 a 2 + a 4 2 a ( a 6 ) + 2 a 2 + 3 a + 4 2 a 2 12 a + a 2 + 2 2 a 2 12 a

4 a 2 + 2 a + 1 a ( a 6 )

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8 x 2 + x 1 x 2 6 x + 8 + 2 x 2 + 3 x x 2 6 x + 8 5 x 2 + 3 x 4 ( x 4 ) ( x 2 )

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

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Fractions with different denominators

Sample set b

Add or Subtract the following rational expressions.

4 a 3 y + 2 a 9 y 2 . The denominators are  n o t  the same . Find the LCD . By inspection, the LCD is 9 y 2 . 9 y 2 + 2 a 9 y 2 The denominator of the first rational expression has been multiplied by 3 y , so the numerator must be multiplied by 3 y . 4 a · 3 y = 12 a y 12 a y 9 y 2 + 2 a 9 y 2 The denominators are now the same . Add the numerators . 12 a y + 2 a 9 y 2

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3 b b + 2 + 5 b b 3 . The denominators are n o t the same . The LCD is ( b + 2 ) ( b 3 ) . ( b + 2 ) ( b 3 ) + ( b + 2 ) ( b 3 ) The denominator of the first rational expression has been multiplied by b 3 , so the numerator must be multiplied by b 3. 3 b ( b 3 ) 3 b ( b 3 ) ( b + 2 ) ( b 3 ) + ( b + 2 ) ( b 3 ) The denominator of the second rational expression has been multiplied by b + 2 , so the numerator must be multiplied by b + 2. 5 b ( b + 2 ) 3 b ( b 3 ) ( b + 2 ) ( b 3 ) + 5 b ( b + 2 ) ( b + 2 ) ( b 3 ) The denominators are now the same . Add the numerators . 3 b ( b 3 ) + 5 b ( b + 2 ) ( b 3 ) ( b + 2 ) = 3 b 2 9 b + 5 b 2 + 10 b ( b 3 ) ( b + 2 ) = 8 b 2 + b ( b 3 ) ( b 2 )

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x + 3 x 1 + x 2 4 x + 4 . The denominators are  n o t  the same . Find the LCD . x + 3 x 1 + x 2 4 ( x + 1 ) The LCD is  ( x + 1 ) ( x 1 ) 4 ( x + 1 ) ( x 1 ) + 4 ( x + 1 ) ( x 1 ) The denominator of the first rational expression has been multiplied by  4 ( x + 1 )  so  the numerator must be multiplied by  4 ( x + 1 ) . 4 ( x + 3 ) ( x + 1 ) 4 ( x + 3 ) ( x + 1 ) 4 ( x + 1 ) ( x 1 ) + 4 ( x + 1 ) ( x 1 ) The denominator of the second rational expression has been multiplied by  x 1 so the numerator must be multiplied by  x 1. ( x 1 ) ( x 2 ) 4 ( x + 3 ) ( x + 1 ) 4 ( x + 1 ) ( x 1 ) + ( x 1 ) ( x 2 ) 4 ( x + 1 ) ( x 1 ) The denominators are now the same. Add the numerators. 4 ( x + 3 ) ( x + 1 ) + ( x 1 ) ( x 2 ) 4 ( x + 1 ) ( x 1 ) 4 ( x 2 + 4 x + 3 ) + x 2 3 x + 2 4 ( x + 1 ) ( x 1 ) 4 x 2 + 16 x + 12 + x 2 3 x + 2 4 ( x + 1 ) ( x 1 ) = 5 x 2 + 13 x + 14 4 ( x + 1 ) ( x 1 )

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x + 5 x 2 7 x + 12 + 3 x 1 x 2 2 x 3 Determine the LCD . x + 5 ( x 4 ) ( x 3 ) + 3 x 1 ( x 3 ) ( x + 1 ) The LCD is ( x 4 ) ( x 3 ) ( x + 1 ) . ( x 4 ) ( x 3 ) ( x + 1 ) + ( x 4 ) ( x 3 ) ( x + 1 ) The first numerator must be multiplied by x + 1 and the second by x 4. ( x + 5 ) ( x + 1 ) ( x 4 ) ( x 3 ) ( x + 1 ) + ( 3 x 1 ) ( x 4 ) ( x 4 ) ( x 3 ) ( x + 1 ) The denominators are now the same . Add the numerators . ( x + 5 ) ( x + 1 ) + ( 3 x 1 ) ( x 4 ) ( x 4 ) ( x 3 ) ( x + 1 ) x 2 + 6 x + 5 + 3 x 2 13 x + 4 ( x 4 ) ( x 3 ) ( x + 1 ) 4 x 2 7 x + 9 ( x 4 ) ( x 3 ) ( x + 1 )

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a + 4 a 2 + 5 a + 6 a 4 a 2 5 a 24 Determine the LCD. a + 4 ( a + 3 ) ( a + 2 ) a 4 ( a + 3 ) ( a 8 ) The LCD is  ( a + 3 ) ( a + 2 ) ( a 8 ) . ( a + 3 ) ( a + 2 ) ( a 8 ) ( a + 3 ) ( a + 2 ) ( a 8 ) The first numerator must be multiplied by  a 8  and the second by  a + 2. ( a + 4 ) ( a 8 ) ( a + 3 ) ( a + 2 ) ( a 8 ) ( a 4 ) ( a + 2 ) ( a + 3 ) ( a + 2 ) ( a 8 ) The denominators are now the same . Subtract the numerators .  ( a + 4 ) ( a 8 ) ( a 4 ) ( a + 2 ) ( a + 3 ) ( a + 2 ) ( a 8 ) a 2 4 a 32 ( a 2 2 a 8 ) ( a + 3 ) ( a + 2 ) ( a 8 ) a 2 4 a 32 a 2 + 2 a + 8 ( a + 3 ) ( a + 2 ) ( a 8 ) 2 a 24 ( a + 3 ) ( a + 2 ) ( a 8 ) Factor  2  from the numerator .  2 ( a + 12 ) ( a + 3 ) ( a + 2 ) ( a 8 )

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3 x 7 x + 5 x x 7 . The denominators are n e a r l y the same.  They differ only in sign. Our technique is to factor  1  from one of them. 3 x 7 x = 3 x ( x 7 ) = 3 x x 7 Factor  1  from the first term .  3 x 7 x + 5 x x 7 = 3 x x 7 + 5 x x 7 = 3 x + 5 x x 7 = 2 x x 7

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Practice set b

Add or Subtract the following rational expressions.

3 x 4 a 2 + 5 x 12 a 3

9 a x + 5 x 12 a 3

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5 b b + 1 + 3 b b 2

8 b 2 7 b ( b + 1 ) ( b 2 )

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a 7 a + 2 + a 2 a + 3

2 a 2 4 a 25 ( a + 2 ) ( a + 3 )

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4 x + 1 x + 3 x + 5 x 3

3 x 2 19 x 18 ( x + 3 ) ( x 3 )

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2 y 3 y + 3 y + 1 y + 4

5 y 2 + 6 y 12 y ( y + 4 )

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a 7 a 2 3 a + 2 + a + 2 a 2 6 a + 8

2 a 2 10 a + 26 ( a 2 ) ( a 1 ) ( a 4 )

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6 b 2 + 6 b + 9 2 b 2 + 4 b + 4

4 b 2 + 12 b + 6 ( b + 3 ) 2 ( b + 2 ) 2

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x x + 4 x 2 3 x 3

2 x 2 5 x + 8 3 ( x + 4 ) ( x 1 )

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5 x 4 x + 7 x x 4

2 x x 4

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Sample set c

Combine the following rational expressions.

3 + 7 x 1 . Rewrite the expression. 3 1 + 7 x 1 The LCD is  x 1. 3 ( x 1 ) x 1 + 7 x 1 = 3 x 3 x 1 + 7 x 1 = 3 x 3 + 7 x 1 = 3 x + 4 x 1

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3 y + 4 y 2 y + 3 y 6 . Rewrite the expression. 3 y + 4 1 y 2 y + 3 y 6 The LCD is  y 6. ( 3 y + 4 ) ( y 6 ) y 6 y 2 y + 3 y 6 = ( 3 y + 4 ) ( y 6 ) ( y 2 y + 3 ) y 6 = 3 y 2 14 y 24 y 2 + y 3 y 6 = 2 y 2 13 y 27 y 6

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Practice set c

Simplify 8 + 3 x 6 .

8 x 45 x 6

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Simplify 2 a 5 a 2 + 2 a 1 a + 3 .

a 2 a 14 a + 3

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Exercises

For the following problems, add or subtract the rational expressions.

6 y 5 x + 8 y 5 x

14 y 5 x

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15 n 2 m 6 n 2 m

9 n 2 m

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y + 4 y 6 + y + 8 y 6

2 y + 12 y 6

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y 1 y + 4 + y + 7 y + 4

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a + 6 a 1 + 3 a + 5 a 1

4 a + 11 a 1

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5 a + 1 a + 7 + 2 a 6 a + 7

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x + 1 5 x + x + 3 5 x

2 x + 4 5 x

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a 6 a + 2 + a 2 a + 2

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b + 1 b 3 + b + 2 b 3

2 b + 3 b 3

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a + 2 a 5 a + 3 a 5

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b + 7 b 6 b 1 b 6

8 b 6

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2 b + 3 b + 1 b 4 b + 1

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3 y + 4 y + 8 2 y 5 y + 8

y + 9 y + 8

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2 a 7 a 9 + 3 a + 5 a 9

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8 x 1 x + 2 15 x + 7 x + 2

7 x 8 x + 2

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2 3 x + 4 6 x 2

2 ( x + 1 ) 3 x 2

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2 5 a 2 1 10 a 3

4 a 1 10 a 3

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4 x 6 + 1 x 1

5 ( x 2 ) ( x 6 ) ( x 1 )

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6 y y + 4 + 2 y y + 3

2 y ( 4 y + 13 ) ( y + 4 ) ( y + 3 )

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x 1 x 3 + x + 4 x 4

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x + 2 x 5 + x 1 x + 2

2 x 2 2 x + 9 ( x 5 ) ( x + 2 )

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a + 3 a 3 a + 2 a 2

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y + 1 y 1 y + 4 y 4

6 y ( y 1 ) ( y 4 )

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

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y + 2 ( y + 1 ) ( y + 6 ) + y 2 y + 6

y 2 ( y + 1 ) ( y + 6 )

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2 a + 1 ( a + 3 ) ( a 3 ) a + 2 a + 3

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3 a + 5 ( a + 4 ) ( a 1 ) 2 a 1 a 1

2 a 2 4 a + 9 ( a + 4 ) ( a 1 )

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2 x x 2 3 x + 2 + 3 x 2

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4 a a 2 2 a 3 + 3 a + 1

7 a 9 ( a + 1 ) ( a 3 )

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3 y y 2 7 y + 12 y 2 y 3

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x 1 x 2 + 6 x + 8 + x + 3 x 2 + 2 x 8

2 ( x 2 + x + 4 ) ( x + 2 ) ( x 2 ) ( x + 4 )

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a 4 a 2 + 2 a 3 + a + 2 a 2 + 3 a 4

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b 3 b 2 + 9 b + 20 + b + 4 b 2 + b 12

2 b 2 + 3 b + 29 ( b 3 ) ( b + 4 ) ( b + 5 )

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y 1 y 2 + 4 y 12 y + 3 y 2 + 6 y 16

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x + 3 x 2 + 9 x + 14 x 5 x 2 4

x + 29 ( x 2 ) ( x + 2 ) ( x + 7 )

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x 1 x 2 4 x + 3 + x + 3 x 2 5 x + 6 + 2 x x 2 3 x + 2

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4 x x 2 + 6 x + 8 + 3 x 2 + x 6 + x 1 x 2 + x 12

5 x 4 3 x 3 34 x 2 + 34 x 60 ( x 2 ) ( x + 2 ) ( x 3 ) ( x + 3 ) ( x + 4 )

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y + 2 y 2 1 + y 3 y 2 3 y 4 y + 3 y 2 5 y + 4

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a 2 a 2 9 a + 18 + a 2 a 2 4 a 12 a 2 a 2 a 6

( a + 5 ) ( a 2 ) ( a + 2 ) ( a 3 ) ( a 6 )

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y 2 y 2 + 6 y + y + 4 y 2 + 5 y 6

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a + 1 a 3 + 3 a 2 a + 6 a 2 a

a 3 8 a 2 18 a 1 a 2 ( a + 3 ) ( a 1 )

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4 3 b 2 12 b 2 6 b 2 6 b

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3 2 x 5 4 x 4 + 2 8 x 3 + 24 x 2

x 3 + 2 x 2 + 6 x + 18 4 x 4 ( x 2 ) ( x + 3 )

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x + 2 12 x 3 + x + 1 4 x 2 + 8 x 12 x + 3 16 x 2 32 x + 16

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2 x x 2 9 x + 1 4 x 2 12 x x 4 8 x 3

14 x 4 9 x 3 2 x 2 + 9 x 36 8 x 3 ( x + 3 ) ( x 3 )

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8 + 2 x + 6

8 x + 50 x + 6

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3 + 5 x 6

3 x 13 x 6

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1 + 3 a a 1

2 a + 1 a 1

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2 x + x 2 4 x + 1

3 x 2 + 2 x 4 x + 1

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3 y + 4 y 2 + 2 y 5 y + 3

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x + 2 + x 2 + 4 x 1

2 x 2 + x + 2 x 1

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3 x 1 x 4 8

5 x + 31 x 4

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2 y 2 + 11 y 1 y + 4 3 y

( y 2 + y + 1 ) y + 4

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5 y 2 2 y + 1 y 2 + y 6 2

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4 a 3 + 2 a 2 + a 1 a 2 + 11 a + 28 + 3 a

7 a 3 + 35 a 2 + 85 a 1 ( a + 7 ) ( a + 4 )

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2 x 1 x + 6 x x 1

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5 m 6 m + 3 m m 6

2 m m 6

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a + 7 8 3 a + 2 a + 1 3 a 8

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2 y + 4 4 5 y 9 5 y 4

2 y 13 5 y 4

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m 1 1 m 2 m 1

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Exercises for review

( [link] ) Simplify ( x 3 y 2 z 5 ) 6 ( x 2 y z ) 2 .

x 22 y 14 z 32

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( [link] ) Write 6 a 3 b 4 c 2 a 1 b 5 c 3 so that only positive exponents appear.

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( [link] ) Construct the graph of y = 2 x + 4.
An xy coordinate plane with gridlines, labeled negative five to five on both axes.

A graph of a line passing through three points with coordinates zero, four; one, two; and two, zero.

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( [link] ) Find the product: x 2 3 x 4 x 2 + 6 x + 5 · x 2 + 5 x + 6 x 2 2 x 8 .

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( [link] ) Replace N with the proper quantity: x + 3 x 5 = N x 2 7 x + 10 .

( x + 3 ) ( x 2 )

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Questions & Answers

explain and give four Example hyperbolic function
Lukman Reply
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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