Question

Solve the rational equation 4 divided by x equals quantity 3 times x plus 2 end quantity divided by x squared, and check for extraneous solutions.

Answer 1

x = 2

Explanation:

Step 1: Write equation

`(4)/(x)=(3x+2)/(x^2)`

Step 2: Solve for x

1. Cross multiply:
   `4x^2=x(3x+2)`
2. Distribute x:
   `4x^2=3x^2+2x`
3. Subtract 3x² on both sides:
   `x^2=2x`
4. Subtract 2x on both sides:
   `x^2-2x=0`
5. Factor:
   `x(x-2)=0`
6. Solve:
   `x = 0, 2`

Step 3: Check

Plug in x to verify it's a solution.

1. Substitute:
   `(4)/(0)=(3(0)+2)/(0^2)`
2. Evaluate: undef (cannot divide by 0)

1. Substitute:
   `(4)/(2)=(3(2)+2)/(2^2)`
2. Evaluate:
   `2=(6+2)/(4)`
3. Evaluate:
   `2=(8)/(4)`
4. Evaluate:
   `2=2`