QuestionSolve the equation: $\frac{3}{x+1}-\frac{1}{2}=\frac{2}{3x+3}$ . Enter solutions as integers or reduced fractions.

Studdy Solution

STEP 1

Assumptions1. The equation is a rational equation, which means it involves fractions. . We are asked to solve for x.
3. The solutions can be in the form of integers or reduced fractions.
4. If there are multiple solutions, they should be separated by commas.

STEP 2

First, we need to get rid of the denominators in the equation. We can do this by finding the least common multiple (LCM) of the denominators and multiplying each term by this LCM.
The denominators are $x+1$ , $2$ , and $x+$ .

STEP 3

We can simplify the third denominator by factoring out a $3$ $3x+3 =3(x+1)$ So, the denominators are $x+1$ , $2$ , and $3(x+1)$ .

STEP 4

The least common multiple (LCM) of these denominators is $6(x+1)$ .Multiply each term of the equation by $6(x+1)$ $6(x+1)\cdot\frac{3}{x+1}-6(x+1)\cdot\frac{1}{2}=6(x+1)\cdot\frac{2}{3x+3}$

STEP 5

implify each term $\cdot3-3(x+1)=2(x+1)$

STEP 6

implify further $18-3x-3=2x+2$

STEP 7

Combine like terms $15-3x=2x+2$

STEP 8

Rearrange the equation to isolate $x$ terms on one side and constant terms on the other side $15-2=3x+2x$

STEP 9

implify further $13=5x$

STEP 10

Finally, solve for $x$ by dividing both sides of the equation by $5$ $x=\frac{13}{5}$ The solution to the equation is $x=\frac{13}{5}$ .

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QuestionSolve the equation: $\frac{3}{x+1}-\frac{1}{2}=\frac{2}{3x+3}$ . Enter solutions as integers or reduced fractions.

Studdy Solution

STEP 1

Assumptions1. The equation is a rational equation, which means it involves fractions. . We are asked to solve for x.
3. The solutions can be in the form of integers or reduced fractions.
4. If there are multiple solutions, they should be separated by commas.

STEP 2

First, we need to get rid of the denominators in the equation. We can do this by finding the least common multiple (LCM) of the denominators and multiplying each term by this LCM.
The denominators are $x+1$ , $2$ , and $x+$ .

STEP 3

We can simplify the third denominator by factoring out a $3$ $3x+3 =3(x+1)$ So, the denominators are $x+1$ , $2$ , and $3(x+1)$ .

STEP 4

The least common multiple (LCM) of these denominators is $6(x+1)$ .Multiply each term of the equation by $6(x+1)$ $6(x+1)\cdot\frac{3}{x+1}-6(x+1)\cdot\frac{1}{2}=6(x+1)\cdot\frac{2}{3x+3}$

STEP 5

implify each term $\cdot3-3(x+1)=2(x+1)$

STEP 6

implify further $18-3x-3=2x+2$

STEP 7

Combine like terms $15-3x=2x+2$

STEP 8

Rearrange the equation to isolate $x$ terms on one side and constant terms on the other side $15-2=3x+2x$

STEP 9

implify further $13=5x$

STEP 10

Finally, solve for $x$ by dividing both sides of the equation by $5$ $x=\frac{13}{5}$ The solution to the equation is $x=\frac{13}{5}$ .

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QuestionSolve the equation: 3x+1−12=23x+3\frac{3}{x+1}-\frac{1}{2}=\frac{2}{3x+3}x+13​−21​=3x+32​. Enter solutions as integers or reduced fractions.

Studdy Solution

STEP 1

Assumptions1. The equation is a rational equation, which means it involves fractions. . We are asked to solve for x.3. The solutions can be in the form of integers or reduced fractions.4. If there are multiple solutions, they should be separated by commas.

STEP 2

First, we need to get rid of the denominators in the equation. We can do this by finding the least common multiple (LCM) of the denominators and multiplying each term by this LCM.The denominators are x+1x+1x+1, 222, and x+x+x+.

STEP 3

We can simplify the third denominator by factoring out a 3333x+3=3(x+1)3x+3 =3(x+1)3x+3=3(x+1)So, the denominators are x+1x+1x+1, 222, and 3(x+1)3(x+1)3(x+1).

STEP 4

The least common multiple (LCM) of these denominators is 6(x+1)6(x+1)6(x+1).Multiply each term of the equation by 6(x+1)6(x+1)6(x+1)6(x+1)⋅3x+1−6(x+1)⋅12=6(x+1)⋅23x+36(x+1)\cdot\frac{3}{x+1}-6(x+1)\cdot\frac{1}{2}=6(x+1)\cdot\frac{2}{3x+3}6(x+1)⋅x+13​−6(x+1)⋅21​=6(x+1)⋅3x+32​

STEP 5

implify each term⋅3−3(x+1)=2(x+1)\cdot3-3(x+1)=2(x+1)⋅3−3(x+1)=2(x+1)

STEP 6

implify further18−3x−3=2x+218-3x-3=2x+218−3x−3=2x+2

STEP 7

Combine like terms15−3x=2x+215-3x=2x+215−3x=2x+2

STEP 8

Rearrange the equation to isolate xxx terms on one side and constant terms on the other side15−2=3x+2x15-2=3x+2x15−2=3x+2x

STEP 9

implify further13=5x13=5x13=5x

STEP 10

Finally, solve for xxx by dividing both sides of the equation by 555x=135x=\frac{13}{5}x=513​The solution to the equation is x=135x=\frac{13}{5}x=513​.

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QuestionSolve the equation: $\frac{3}{x+1}-\frac{1}{2}=\frac{2}{3x+3}$ . Enter solutions as integers or reduced fractions.

Assumptions1. The equation is a rational equation, which means it involves fractions. . We are asked to solve for x.
3. The solutions can be in the form of integers or reduced fractions.
4. If there are multiple solutions, they should be separated by commas.

First, we need to get rid of the denominators in the equation. We can do this by finding the least common multiple (LCM) of the denominators and multiplying each term by this LCM.
The denominators are $x+1$ , $2$ , and $x+$ .

We can simplify the third denominator by factoring out a $3$ $3x+3 =3(x+1)$ So, the denominators are $x+1$ , $2$ , and $3(x+1)$ .

The least common multiple (LCM) of these denominators is $6(x+1)$ .Multiply each term of the equation by $6(x+1)$ $6(x+1)\cdot\frac{3}{x+1}-6(x+1)\cdot\frac{1}{2}=6(x+1)\cdot\frac{2}{3x+3}$

implify each term $\cdot3-3(x+1)=2(x+1)$

implify further $18-3x-3=2x+2$

Combine like terms $15-3x=2x+2$

Rearrange the equation to isolate $x$ terms on one side and constant terms on the other side $15-2=3x+2x$

implify further $13=5x$

Finally, solve for $x$ by dividing both sides of the equation by $5$ $x=\frac{13}{5}$ The solution to the equation is $x=\frac{13}{5}$ .