What would be an extraneous solution to the equation 4/(x+7) +3 = -2/13
4/(x+7) +3 = -2/13
4/(x+7) = -2/13 - 3
4/(x+7) = -41/13
52 = -41x - 287
41x = -339
x = -339/41 <----- noting extraneous about that
To find the extraneous solutions to the equation 4/(x+7) + 3 = -2/13, we need to solve the equation and then check if any of the solutions make the equation undefined.
Step 1: Solve the equation
First, let's simplify the equation by getting rid of the fractions. Multiply every term by (x+7) to eliminate the denominators:
4 * (x+7)/(x+7) + 3 * (x+7) = -2/13 * (x+7)
This simplifies to:
4 + 3(x+7) = -2/13 * (x+7)
Expanding and simplifying:
4 + 3x + 21 = (-2/13)x - (2/13) * 7
4 + 3x + 21 = (-2/13)x - 14/13
Combining like terms:
3x + 25 = (-2/13)x - 14/13
Next, let's eliminate the fractions by multiplying every term by 13:
13 * (3x + 25) = 13 * ((-2/13)x - 14/13)
This simplifies to:
39x + 325 = -2x - 14
Combine like terms:
39x + 2x = -14 - 325
41x = -339
Divide by 41 to solve for x:
x = -339/41
So, the solution to the equation is x = -339/41.
Step 2: Check for extraneous solutions
To check if there are any extraneous solutions, substitute the value of x back into the original equation 4/(x+7) + 3 = -2/13:
4/((-339/41)+7) + 3 = -2/13
Simplifying the expression in the denominator:
4/(4/41) + 3 = -2/13
Multiplying by the reciprocal:
4 * (41/4) + 3 = -2/13
Simplifying the expression:
41 + 3 = -2/13
44 = -2/13
Since 44 is not equal to -2/13, there are no extraneous solutions to the equation 4/(x+7) + 3 = -2/13.
Therefore, the solution x = -339/41 is the valid solution to the equation.