Solve for x x/(2x+7)-(x+1)/(x+3)=1

Problem

x/(2*x+7)−(x+1)/(x+3)=1

Solution

1. Identify the common denominator by multiplying the denominators of the fractions on the left side, which is (2*x+7)*(x+3)

2. Multiply the entire equation by the common denominator (2*x+7)*(x+3) to eliminate the fractions.

x*(x+3)−(x+1)*(2*x+7)=1*(2*x+7)*(x+3)

3. Expand the terms on both sides of the equation using the distributive property.

(x2+3*x)−(2*x2+7*x+2*x+7)=2*x2+6*x+7*x+21

4. Simplify the expressions by combining like terms within the parentheses and on the right side.

x2+3*x−(2*x2+9*x+7)=2*x2+13*x+21

5. Distribute the negative sign and combine terms on the left side.

x2+3*x−2*x2−9*x−7=2*x2+13*x+21

−x2−6*x−7=2*x2+13*x+21

6. Rearrange into a quadratic equation by moving all terms to one side to set the equation to zero.

0=3*x2+19*x+28

7. Factor the quadratic equation by finding two numbers that multiply to 3×28=84 and add to 19 These numbers are 7 and 12

3*x2+12*x+7*x+28=0

3*x*(x+4)+7*(x+4)=0

(3*x+7)*(x+4)=0

8. Solve for x by setting each factor equal to zero.

3*x+7=0⇒x=−7/3

x+4=0⇒x=−4

9. Check for extraneous solutions by ensuring the denominators 2*x+7 and x+3 are not zero at these values. Both values are valid.

Final Answer

x/(2*x+7)−(x+1)/(x+3)=1⇒x=−7/3,−4

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