How do you solve #1 /( 4 x + 2) = 5# and find any extraneous solutions?

Question

Genesis Allen · Accepted Answer

#x=-9/20#

Multiply both sides by the fraction's denominator, (4x + 2), to eliminate the fraction on the left side of the equation.

#rArrcancel((4x+2))xx1/cancel((4x+2))=5(4x+2)#

#rArr1=20x+10#

Deduct 10 from each side.

#1-10=20xcancel(+10)cancel(-10)#

#rArr20x=-9#

Divide both sides by 20 to find x.

#(cancel(20) x)/cancel(20)=(-9)/20#

#rArrx=-9/20" is the only solution"#

Eli Assouline · Answer

undefined To solve the equation 1/(4x + 2) = 5 and identify any extraneous solutions, follow these steps:

1. Multiply both sides of the equation by (4x + 2) to eliminate the denominator:
   1 = 5(4x + 2)

2. Distribute 5 to the terms inside the parentheses:
   1 = 20x + 10

3. Subtract 10 from both sides of the equation:
   -9 = 20x

4. Divide both sides by 20 to isolate x:
   x = -9/20

5. To check for extraneous solutions, substitute the found value of x back into the original equation:
   1/(4(-9/20) + 2) = 5

6. Simplify the expression inside the parentheses:
   1/(-9/5 + 2) = 5

7. Further simplify:
   1/(-9/5 + 10/5) = 5
   1/(1/5) = 5
   5 = 5

8. Since the equation is true, the solution x = -9/20 is not extraneous.

Therefore, the solution to the equation 1/(4x + 2) = 5 is x = -9/20, and there are no extraneous solutions.