Solve. Check your answer.

1. ((3x-7)/(x-5)) + ((x)/(2)) = ((8)/(x-5))

A: ?

{ (6x-14) + (x^2-5x) ] /[2(x-5) ] =16/[2(x-5)]

x^2 + x - 14 = 16

x^2 + x - 30 = 0

(x-5)(x+6) = 0

x = 5 or -6

Oh delete the x = 5, denominator of original is 0

Not in domain

Nasty !!! at 2:30 m

To solve this equation, we will first simplify both sides of the equation. Then, we will find a common denominator and combine the fractions. Finally, we will isolate the variable x and solve for its value.

Here are the steps to solve the equation:

1. Simplify both sides of the equation:

((3x-7)/(x-5)) + (x/2) = 8/(x-5)

2. Find a common denominator for the fractions on the left side of the equation. The least common denominator (LCD) for (x-5) and 2 is 2(x-5).

Multiply the fraction (3x-7)/(x-5) by 2/2 to get a common denominator:

((3x-7)/(x-5)) * (2/2) = 2(3x-7)/(2(x-5))

Now, the equation becomes:

(2(3x-7)/(2(x-5))) + (x/2) = 8/(x-5)

3. Combine the fractions on the left side of the equation:

((6x-14)/(2(x-5))) + (x/2) = 8/(x-5)

4. Find a common denominator for the fractions on the left side. The LCD for 2(x-5) and 2 is 2(x-5).

Multiply the fraction (x/2) by (x-5)/(x-5) to get a common denominator:

((6x-14)/(2(x-5))) + ((x(x-5))/(2(x-5))) = 8/(x-5)

Now, the equation becomes:

((6x-14)/(2(x-5))) + (x(x-5))/(2(x-5)) = 8/(x-5)

5. Combine the fractions on the left side of the equation:

((6x-14)+(x(x-5)))/(2(x-5)) = 8/(x-5)

6. Simplify the expression on the left side of the equation:

(6x-14+x^2-5x)/(2(x-5)) = 8/(x-5)

Now, the equation becomes:

(x^2+x-14)/(2(x-5)) = 8/(x-5)

7. Multiply both sides of the equation by 2(x-5) to eliminate the denominators:

2(x-5) * (x^2+x-14)/(2(x-5)) = (8/(x-5)) * 2(x-5)

Cancel out the common factors:

x^2+x-14 = 8

8. Rearrange the equation to have zero on one side:

x^2+x-14 - 8 = 0

9. Simplify the equation:

x^2+x-22 = 0

10. Solve the quadratic equation by factoring, completing the square, or using the quadratic formula.

Using the quadratic formula, we have:

x = (-b ± √(b^2 - 4ac))/(2a)

In this case, a = 1, b = 1, and c = -22.

x = (-(1) ± √((1)^2 - 4(1)(-22)))/(2(1))

Simplifying further:

x = (-1 ± √(1 + 88))/2

x = (-1 ± √89)/2

So, the possible values for x are:

x = (-1 + √89)/2 or x = (-1 - √89)/2

To check your answer, substitute the values of x in the original equation and see if both sides are equal.