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.