How do you solve #(6x)/(x+4)+4=(2x+2)/(x-1)#?

Question

Eva Abramson · Accepted Answer

The solution for this equation is #x =2,-3/2#

#(6x)/(x+4)+4=(2x+2)/(x-1)#

or, #(6x+4*(x+4))/(x+4)=(2x+2)/(x-1) rarr# Take LCM on LHS.

or, #(6x+4x+16)/(x+4)=(2x+2)/(x-1) #

or, #(10x+16)/(x+4)=(2x+2)/(x-1) #

or, #(x-1)(10x+16)=(x+4)(2x+2) rarr# Cross Multiply

or, #x(10x+16)-1(10x+16)=x(2x+2)+4(2x+2) rarr#Simplify

or, #10x^2+16x-10x-16=2x^2+2x+8x+8rarr#Simplify

or, #10x^2+6x-16=2x^2+10x+8#

or, #2(5x^2+3x-8)=2(x^2+5x+4)rarr#Take 2 Common on both LHS Side and RHS Side

or, #(2(5x^2+3x-8))/2=(2(x^2+5x+4))/2rarr#Dividing both sides by 2

or, #5x^2+3x-8=x^2+5x+4#

or, #5x^2-x^2+3x-3x-8=x^2-x^2+5x-3x+4rarr#Subracting #x^2# and 3x from boths sides

or, #4x^2-8=2x+4#

or, #4x^2-8-4=2x+4-4rarr#Subtracting 4 from both sides

or, #4x^2-12=2x#

or, #4x^2-2x-12=2x-2xrarr# Subtracting 2x from both sides

or, #4x^2-2x-12=0#

or, #4x^2-8x+6x-12=0rarr# Factoring LHS

or, #4x(x-2)+6(x-2)=0#

or, #(4x+6)(x-2)=0#

Either 4x+6=0 or, #4x=-6#

or, #x=-(6/4)#

or, #x=-(3/2)#

Or,

#x-2=0#

or, #x=2#

:.The solution for #x=2,-3/2#.

Nathan Axel · Answer

undefined To solve the equation (6x)/(x+4)+4=(2x+2)/(x-1), you can follow these steps:

1. Start by multiplying both sides of the equation by the denominators (x+4) and (x-1) to eliminate the fractions.

2. Simplify the equation by distributing and combining like terms.

3. Rearrange the equation to isolate the variable on one side.

4. Solve for x by applying the appropriate algebraic operations.

The solution to the equation will be the value(s) of x that satisfy the equation after simplification and solving.

Savannah Anderson · Answer

undefined To solve the equation (6x)/(x+4) + 4 = (2x+2)/(x-1), follow these steps:

1. Find a common denominator for both fractions.
2. Multiply both sides of the equation by the common denominator to clear the fractions.
3. Simplify and solve the resulting equation.
4. Check for any extraneous solutions.

Here's a step-by-step breakdown:

1. The common denominator for the fractions is (x+4)(x-1).
2. Multiply both sides of the equation by (x+4)(x-1) to clear the fractions:

(x+4)(x-1) * [(6x)/(x+4) + 4] = (x+4)(x-1) * [(2x+2)/(x-1)]

3. Simplify:

6x(x-1) + 4(x+4)(x-1) = (2x+2)(x+4)

4. Expand and simplify each side:

6x^2 - 6x + 4(x^2 + 3x - 4) = 2x^2 + 10x + 8

6x^2 - 6x + 4x^2 + 12x - 16 = 2x^2 + 10x + 8

10x^2 + 6x - 16 = 2x^2 + 10x + 8

10x^2 - 2x^2 + 6x - 10x - 16 - 8 = 0

8x^2 - 4x - 24 = 0

5. Factor or use the quadratic formula to solve for x.

6. After solving for x, check the solutions to ensure they are valid for the original equation.

That's the method to solve the given equation.