To eliminate the fractions, we can multiply every term by x(x + 3):
(x - 3) - x(x - 5) = x(x + 3) * 1 - (x - 5)
Expanding and simplifying:
x - 3 - x^2 + 5x = x^2 + 3x - (x - 5)
Combining like terms:
6x - 3 - x^2 = x^2 + 3x - x + 5
Moving all terms to one side:
x^2 + 6x + x - 3 - 3x - 3x - 5 = 0
Simplifying:
x^2 + x - 11 = 0
Using the quadratic formula, we have:
x = (-b ± √(b^2 - 4ac))/(2a)
Plugging in the values a = 1, b = 1, and c = -11:
x = (-1 ± √(1^2 - 4(1)(-11)))/(2(1))
Simplifying:
x = (-1 ± √(1 + 44))/2
x = (-1 ± √45)/2
So the solutions to the equation are x = (-1 + √45)/2 and x = (-1 - √45)/2.
g. To solve the equation x/(2x + 7) - 1 = (x + 1)/(x + 3), we can start by multiplying every term by the common denominator (2x + 7)(x + 3) to eliminate the fractions:
x(x + 3) - (2x + 7)(x + 3) = (x + 1)(2x + 7)
Expanding the terms:
x^2 + 3x - (2x^2 + 13x + 21) = 2x^2 + 9x + 7
Simplifying:
x^2 + 3x - 2x^2 - 13x - 21 = 2x^2 + 9x + 7
Rearranging terms and combining like terms:
-x^2 - 23x - 21 = 2x^2 + 9x + 7
Moving all terms to one side:
3x^2 + 32x + 28 = 0
This quadratic equation does not factor easily, so we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Plugging in the values a = 3, b = 32, and c = 28:
x = (-32 ± √(32^2 - 4(3)(28)))/(2(3))
Simplifying:
x = (-32 ± √(1024 - 336))/6
x = (-32 ± √688)/6
So the solutions to the equation are x = (-32 + √688)/6 and x = (-32 - √688)/6.
h. To solve the equation (x + 1)/(4x + 5) + (2x - 3)/(2x - 7) = 1/3, we can start by finding a common denominator for the fractions. In this case, the common denominator is (4x + 5)(2x - 7)(3):
Answers & Comments
Answer:
a. To solve the equation 6/x + 3/(x^2) = 9/x, we can multiply every term by x to eliminate the denominators:
6 + 3/x = 9
Now, we can subtract 6 from both sides:
3/x = 3
Next, we can multiply both sides by x:
3 = 3x
Finally, we divide both sides by 3:
x = 1
So the solution to the equation is x = 1.
b. To solve the equation -2 + x = 6/x + 3, we can start by moving all terms to one side:
x - 6/x = 2 + 3
Next, we can combine the terms on the right side:
x - 6/x = 5
To eliminate the fraction, we can multiply every term by x:
x^2 - 6 = 5x
Now, we can rearrange the equation and set it equal to zero:
x^2 - 5x - 6 = 0
Factoring the quadratic equation, we have:
(x - 6)(x + 1) = 0
Setting each factor equal to zero and solving for x, we get:
x - 6 = 0 --> x = 6
x + 1 = 0 --> x = -1
So the solutions to the equation are x = 6 and x = -1.
c. To solve the equation 2/(x^2 - 9) - 4/(x - 3) = 5/(x + 3), we first need to factor the denominators:
x^2 - 9 = (x - 3)(x + 3)
Now, we can rewrite the equation with the factored denominators:
2/\[(x - 3)(x + 3)\] - 4/(x - 3) = 5/(x + 3)
To eliminate the fractions, we can multiply every term by (x - 3)(x + 3):
2 - 4(x + 3) = 5(x - 3)
Simplifying:
2 - 4x - 12 = 5x - 15
Combining like terms:
-4x - 10 = 5x - 15
Moving all terms to one side:
9x = 5
Dividing both sides by 9:
x = 5/9
So the solution to the equation is x = 5/9.
d. To solve the equation x/(x - 3) = (x + 1)/(-4), we can cross-multiply:
x * (-4) = (x - 3) * (x + 1)
Simplifying:
-4x = x^2 - 2x - 3
Moving all terms to one side:
x^2 - 2x - 3 + 4x = 0
Combining like terms:
x^2 + 2x - 3 = 0
Factoring the quadratic equation, we have:
(x - 1)(x + 3) = 0
Setting each factor equal to zero and solving for x, we get:
x - 1 = 0 --> x = 1
x + 3 = 0 --> x = -3
So the solutions to the equation are x = 1 and x = -3.
e. To solve the equation 3/(x + 1) - x/(x - 1) = -11/(x^2 - 1), we can start by factoring the denominator on the right side:
x^2 - 1 = (x - 1)(x + 1)
Now, we can rewrite the equation with the factored denominator:
3/(x + 1) - x/(x - 1) = -11/\[(x - 1)(x + 1)\]
To eliminate the fractions, we can multiply every term by (x - 1)(x + 1):
3(x - 1) - x(x + 1) = -11
Simplifying:
3x - 3 - x^2 - x = -11
Rearranging terms and combining like terms:
x^2 + 4x - 8 = 0
Using the quadratic formula, we have:
x = (-b ± √(b^2 - 4ac))/(2a)
Plugging in the values a = 1, b = 4, and c = -8:
x = (-4 ± √(4^2 - 4(1)(-8)))/(2(1))
Simplifying:
x = (-4 ± √(16 + 32))/2
x = (-4 ± √48)/2
x = (-4 ± 4Continuing from the previous response:
Taking the square root:
x = (-4 ± 4√3)/2
Simplifying further:
x = -2 ± 2√3
So the solutions to the equation are x = -2 + 2√3 and x = -2 - 2√3.
f. To solve the equation (x - 3)/(x^2 + 3x) = 1/(x + 3) - (x - 5)/(x^2 + 3x), we can start by simplifying the denominators:
x^2 + 3x = x(x + 3)
x^2 + 3x = (x + 3)(x)
Now, we can rewrite the equation with the simplified denominators:
(x - 3)/(x(x + 3)) = 1/(x + 3) - (x - 5)/(x(x + 3))
To eliminate the fractions, we can multiply every term by x(x + 3):
(x - 3) - x(x - 5) = x(x + 3) * 1 - (x - 5)
Expanding and simplifying:
x - 3 - x^2 + 5x = x^2 + 3x - (x - 5)
Combining like terms:
6x - 3 - x^2 = x^2 + 3x - x + 5
Moving all terms to one side:
x^2 + 6x + x - 3 - 3x - 3x - 5 = 0
Simplifying:
x^2 + x - 11 = 0
Using the quadratic formula, we have:
x = (-b ± √(b^2 - 4ac))/(2a)
Plugging in the values a = 1, b = 1, and c = -11:
x = (-1 ± √(1^2 - 4(1)(-11)))/(2(1))
Simplifying:
x = (-1 ± √(1 + 44))/2
x = (-1 ± √45)/2
So the solutions to the equation are x = (-1 + √45)/2 and x = (-1 - √45)/2.
g. To solve the equation x/(2x + 7) - 1 = (x + 1)/(x + 3), we can start by multiplying every term by the common denominator (2x + 7)(x + 3) to eliminate the fractions:
x(x + 3) - (2x + 7)(x + 3) = (x + 1)(2x + 7)
Expanding the terms:
x^2 + 3x - (2x^2 + 13x + 21) = 2x^2 + 9x + 7
Simplifying:
x^2 + 3x - 2x^2 - 13x - 21 = 2x^2 + 9x + 7
Rearranging terms and combining like terms:
-x^2 - 23x - 21 = 2x^2 + 9x + 7
Moving all terms to one side:
3x^2 + 32x + 28 = 0
This quadratic equation does not factor easily, so we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Plugging in the values a = 3, b = 32, and c = 28:
x = (-32 ± √(32^2 - 4(3)(28)))/(2(3))
Simplifying:
x = (-32 ± √(1024 - 336))/6
x = (-32 ± √688)/6
So the solutions to the equation are x = (-32 + √688)/6 and x = (-32 - √688)/6.
h. To solve the equation (x + 1)/(4x + 5) + (2x - 3)/(2x - 7) = 1/3, we can start by finding a common denominator for the fractions. In this case, the common denominator is (4x + 5)(2x - 7)(3):
\[(x + 1)(2x - 7) + (2x - 3)(4x + 5)\]/(4x + 5)(2x - 7) = 1/3
Expanding and simplifying:
\[2x^2 - 3x - 7 + 8x^2 - 6x + 10x - 15\]/\[(4x + 5)(2x - 7)\] = 1/3