Question

Solve the rational equation for x and state all x values that are excluded from the solution set.

If there is more than one excluded value then separate them with a comma and do not include any spaces. If a value is not an integer (such as a fraction) then type it as a decimal rounded to the nearest hundredth.


\frac{5}{x+1}+\frac{1}{x-3}=\frac{-6}{x^2-2x-3}

Solving for x gives us x= ?

The value for x cannot equal ?

Answer 1

The solution to the rational equation is x=1.33, rounding to the nearest hundredth. The excluded values for x, which cannot be included in the solution set because they would make the denominator zero, are -1 and 3.

Step-by-step explanation:

To solve the rational equation \( \frac{5}{x+1}+\frac{1}{x-3}=\frac{-6}{x^2-2x-3} \), we need to find a common denominator to combine the fractions on the left side. The denominator on the right side of the equation factors to (x+1)(x-3), so we will use this for our common denominator.

Multiply both sides of the equation by (x+1)(x-3) to eliminate the denominators:

\( 5(x-3) + 1(x+1) = -6 \)

Simplify and combine like terms:

\( 5x - 15 + x + 1 = -6 \)

\( 6x - 14 = -6 \)

Add 14 to both sides:

\( 6x = 8 \)

Now, divide both sides by 6 to solve for x:

\( x = \frac{8}{6} \)

\( x = \frac{4}{3} \)

So, solving for x gives us x = 1.33 when rounded to the nearest hundredth.

The values for x that cannot be included in the solution set are the values that would make any denominator zero. These values are x = -1 and x = 3. Thus, the excluded values for x are -1,3.