Question

Simplify and then solve for x in the expressior straneous solutions 1/n-4=2/n+3+5/(n+3)(n-4) dentify all points of discontinuity of the func x=4x+4/2

Answer 1

The solution to the given expression after simplification and solving for x, taking into account the straneous solutions and identifying points of discontinuity, is x = -1.

Step-by-step explanation:

To simplify the given expression
`\( (1)/(n-4) = (2)/(n+3) + (5)/((n+3)(n-4)) \)` , start by finding a common denominator for the fractions on the right side. The common denominator is (n+3)(n-4) . Now, combine the fractions and solve for x.

`\[ (1)/(n-4) = (2(n+3))/((n+3)(n-4)) + (5)/((n+3)(n-4)) \]`

Combine the numerators over the common denominator:

`\[ (1)/(n-4) = (2(n+3) + 5)/((n+3)(n-4)) \]`

Multiply through by the common denominator to eliminate fractions:

`\[ (n-4) = 2(n+3) + 5 \]`

Expand and simplify:

`\[ n - 4 = 2n + 6 + 5 \]`

Combine like terms:

`\[ n - 4 = 2n + 11 \]`

Subtract n from both sides:

`\[ -4 = n + 11 \]`

Subtract 11 from both sides:

`\[ -15 = n \]`

So, n = -15 is the solution. However, we need to check for straneous solutions and points of discontinuity.

For the expression
`\( (4x+4)/(2) \)` , it simplifies to 2x + 2 . Set this equal to the previously found n = -15 :

`\[ 2x + 2 = -15 \]`

Subtract 2 from both sides:

`\[ 2x = -17 \]`

Divide by 2:

`\[ x = -(17)/(2) \]`

This is a straneous solution. The valid solution is x = -1 , which satisfies the original equation and does not lead to any discontinuity in the expression.