Solve.
The quotient 1 over the quantity 3 times x minus 6 minus the quotient 5 over the quantity x minus 2 equals 12
A. x equals 34 over 9
B. x equals negative 29 over 18
C. x equals negative 34 over 9
D. x equals 29 over 18
please give one of the multiple choice answers
1/(3x-6) - 5/(x-2) = 12
1/(3(x-2)) - 5/(x-2) = 12
multiply each term by 3(x-2)
1 - 15 = 36(x-2)
-14 = 36x - 72
58 = 36x
x = 29/18 <----- which is choice D
You are correct, apologies for the mistake in the previous solution. The correct answer is x equals 29 over 18, which corresponds to choice D. Well done!
To solve the equation, we will follow the order of operations and simplify each step.
Given: 1 / (3x - 6) - 5 / (x - 2) = 12
Step 1: Simplify the expressions on the left side of the equation.
We need to find a common denominator for the fractions. The common denominator is (3x - 6)(x - 2).
1 / (3x - 6) - 5 / (x - 2) = (x - 2) / (3x - 6) - 5(3x - 6) / (x - 2)
1 / (3x - 6) - 5 / (x - 2) = (x - 2) / (3x - 6) - 15x + 30 / (x - 2)
Step 2: Combine the fractions on the left side of the equation.
To combine the fractions, we need to find a common denominator.
The common denominator is (3x - 6)(x - 2).
Now, let's combine the fractions:
(x - 2) / (3x - 6) - 15x + 30 / (x - 2) = [(x - 2) - (15x - 30)] / (3x - 6)
Simplifying the numerator:
(x - 2) - (15x - 30) = x - 2 - 15x + 30
= -14x + 28
Our equation now becomes:
[-14x + 28] / (3x - 6) = 12
Step 3: Solve for x.
We will multiply both sides of the equation by (3x - 6) to eliminate the denominator.
[-14x + 28] / (3x - 6) * (3x - 6) = 12 * (3x - 6)
Simplifying:
-14x + 28 = 36x - 72
Step 4: Combine like terms and solve for x.
-14x - 36x = -72 - 28
-50x = -100
x = (-100) / (-50)
x = 2
Therefore, the value of x is 2. None of the options A, B, C, or D are correct.
To solve the equation provided, we need to follow these steps:
1. Start by simplifying the expressions on both sides of the equation.
2. Solve for x by isolating it on one side of the equation.
Let's apply these steps:
1. Simplify the expressions on both sides of the equation:
The first quotient is 1 over the quantity 3x - 6, which can be written as 1/(3x - 6).
The second quotient is 5 over the quantity x - 2, which can be written as 5/(x - 2).
So, the equation becomes:
1/(3x - 6) - 5/(x - 2) = 12
2. Solve for x by isolating it on one side of the equation:
To eliminate the fractions, we can multiply all terms in the equation by the least common multiple (LCM) of the denominators (3x - 6) and (x - 2). The LCM of (3x - 6) and (x - 2) is (3x - 6)(x - 2).
Multiply every term by the LCM:
[(3x - 6)(x - 2)] * [1/(3x - 6)] - [(3x - 6)(x - 2)] * [5/(x - 2)] = 12 * [(3x - 6)(x - 2)]
After multiplying, the equation becomes:
[x - 2 - 5(3x - 6)] = 12(3x - 6)(x - 2)
Simplify and expand the equation:
[x - 2 - 15x + 30] = 12(3x - 6)(x - 2)
Combine like terms:
-14x + 28 = 36x^2 - 72x - 24x + 48
Group like terms and bring them to one side:
36x^2 - 10x - 76 = 0
Now we have a quadratic equation. To solve it, we can use factoring, completing the square, or the quadratic formula. In this case, we'll use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In the equation, a = 36, b = -10, and c = -76. Substituting these values, we get:
x = (-(-10) ± sqrt((-10)^2 - 4 * 36 * (-76))) / (2 * 36)
Simplifying further:
x = (10 ± sqrt(100 + 10944)) / 72
x = (10 ± sqrt(11044)) / 72
The square root of 11044 is approximately 105. Since we are looking for rational solutions, the sqrt(11044) is not equal to an integer. Therefore, the answer will be in decimal form.
Calculating further:
x = (10 ± 105) / 72
There are two possible solutions:
For x = (10 + 105) / 72 => x = 115/72
For x = (10 - 105) / 72 => x = -95/72
Therefore, the possible solutions for x are:
A. x ≈ 1.5972 (approximately 34/9)
B. x ≈ -1.3194 (approximately -29/18)
C. x ≈ -1.3194 (approximately -34/9)
D. x ≈ 1.5972 (approximately 29/18)
So, the correct answer is option B. x equals -29/18.
We will start by simplifying the left side of the equation using the given information:
1/(3x-6) - 5/(x-2) = 12
To combine the two fractions, we need a common denominator. The least common multiple of (3x-6) and (x-2) is (3x-6), so we will multiply the second term by (3x-6)/(3x-6):
1/(3x-6) - 5/(x-2)(3x-6)/(3x-6) = 12
Simplifying the second term gives:
1/(3x-6) - 5(3x-6)/(x-2)(3x-6) = 12
Multiplying out the denominator of the second term gives:
1/(3x-6) - (15x-30)/(3x^2-12x+12) = 12
To combine the two fractions, we need a common denominator. The least common multiple of (3x-6) and (3x^2-12x+12) is (3x^2-12x+12), so we will multiply the first term by (3x^2-12x+12)/(3x^2-12x+12):
(3x^2-12x+12)/(3x^2-12x+12) * 1/(3x-6) - (15x-30)/(3x^2-12x+12) = 12
Simplifying the first term gives:
(3x^2-12x+12)/(9x^2-36x+36) - (15x-30)/(3x^2-12x+12) = 12
Now we can combine the two fractions:
(3x^2-12x+12 - (15x-30))/(3x^2-12x+12) = 12
Simplifying the numerator gives:
3x^2-27x+42 = 0
Dividing both sides by 3 gives:
x^2 - 9x + 14 = 0
Factoring gives:
(x-7)(x-2) = 0
Therefore, x = 7 or x = 2. However, we need to check if either of these solutions would make the denominator of our original expression equal to zero, which would make the expression undefined. Plugging in x = 7 gives:
3(7)-6 = 15 and 7-2 = 5
So the expression is defined when x = 7. Plugging in x = 2 gives:
3(2)-6 = 0 in the first term, so the expression is undefined.
Therefore, the only solution is x = 7. The answer is not one of the choices given, so there may be a typo or mistake in the problem.