solve the rational equation indicate any extraneous solutions 5/x + 8/x+3=7
To solve the rational equation 5/x + 8/(x+3) = 7, we can start by finding a common denominator for the two fractions on the left side. The common denominator is x(x+3), so we can rewrite the equation as:
(5(x+3) + 8x) / x(x+3) = 7
Simplifying the numerator, we have:
(5x + 15 + 8x) / x(x+3) = 7
(13x + 15) / x(x+3) = 7
Next, we can multiply both sides of the equation by the denominator x(x+3) to get rid of the fraction:
13x + 15 = 7x(x+3)
Expanding the right side, we have:
13x + 15 = 7x^2 + 21x
Rearranging the equation to bring all terms to one side, we have:
7x^2 + 21x - 13x - 15 = 0
7x^2 + 8x - 15 = 0
Now, we can factor the quadratic equation:
(7x - 5)(x + 3) = 0
Setting each factor equal to zero, we have two possible solutions:
7x - 5 = 0, which gives x = 5/7
x + 3 = 0, which gives x = -3
Now we need to check if these solutions are valid or if they are extraneous. We substitute each solution back into the original equation:
For x = 5/7:
5/(5/7) + 8/(5/7 + 3) = 7
5 * 7/5 + 8/(35/7 + 3) = 7
7 + 8/(35/7 + 3) = 7
7 + 8/(8 + 3) = 7
7 + 8/11 = 7
7 + 8/11 = 7
This equation is true, so x = 5/7 is a valid solution.
For x = -3:
5/(-3) + 8/(-3 + 3) = 7
-5/3 + 8/0 = 7
The expression 8/0 is undefined, so x = -3 is an extraneous solution.
Therefore, the only valid solution to the original rational equation 5/x + 8/(x+3) = 7 is x = 5/7.
To solve the rational equation 5/x + 8/(x+3) = 7, you can follow these steps:
Step 1: Find a common denominator.
The common denominator for the fractions 5/x and 8/(x+3) is x(x+3). Rewrite the equation with the common denominator:
(5(x+3) + 8x) / (x(x+3)) = 7.
Step 2: Simplify the equation.
Distribute the 5 to (x+3): (5x+15 + 8x) / (x(x+3)) = 7.
Combine like terms: (13x+15) / (x(x+3)) = 7.
Step 3: Remove the fraction by multiplying both sides of the equation by the denominator.
Multiply both sides of the equation by x(x+3):
(x(x+3)) * (13x+15)/(x(x+3)) = 7 * x(x+3).
This simplifies to: 13x + 15 = 7x(x+3).
Step 4: Solve for x.
Expand the right side: 13x + 15 = 7x^2 + 21x.
Move all terms to one side to set the equation to zero:
7x^2 + 8x - 15 = 0.
Step 5: Factor the quadratic equation.
To factor 7x^2 + 8x - 15, find two numbers that multiply to -105 (product of the coefficient of x^2 and the constant term) and add up to the coefficient of x (8 in this case).
The numbers are 15 and -7. Rewrite the equation as:
7x^2 + 15x - 7x - 15 = 0.
Factor by grouping:
(7x^2 + 15x) - (7x + 15) = 0.
Factor out the common terms in each grouping:
7x(x + 3) - 1(x + 3) = 0.
Combine the terms:
(7x - 1)(x + 3) = 0.
Step 6: Set each factor equal to zero and solve for x.
7x - 1 = 0 or x + 3 = 0.
For the first factor, solve for x:
7x - 1 = 0.
Add 1 to both sides: 7x = 1.
Divide both sides by 7: x = 1/7.
For the second factor, solve for x:
x + 3 = 0.
Subtract 3 from both sides: x = -3.
Step 7: Check for extraneous solutions.
Plug in the solutions found, x = 1/7 and x = -3, back into the original equation and see if they result in a true statement:
For x = 1/7:
5/(1/7) + 8/(1/7 + 3) = 7.
This simplifies to 35 + 8/(22/7) = 7, which is not true.
For x = -3:
5/(-3) + 8/(-3 + 3) = 7.
This simplifies to -5/3 + 8/0 = 7, which is not true.
Since neither solution satisfies the original equation, there are no valid solutions for this rational equation.
To solve the rational equation (5/x) + (8/(x + 3)) = 7, you need to follow these steps:
Step 1: Find a common denominator for the fractions on the left side of the equation. In this case, the common denominator is (x)(x + 3).
Step 2: Multiply each term in the equation by the common denominator to eliminate the fractions. Doing so results in: 5(x + 3) + 8(x) = 7(x)(x + 3).
Step 3: Simplify the equation by expanding and combining like terms. The equation becomes: 5x + 15 + 8x = 7x^2 + 21x.
Step 4: Move all the terms to one side of the equation to set it equal to zero. Rearranging, the equation becomes: 0 = 7x^2 + 21x - 5x - 8x - 15.
Step 5: Simplify and rearrange the equation to: 0 = 7x^2 + 8x - 15.
Step 6: Factor the quadratic equation. In this case, the factors are (7x + 15)(x - 1) = 0.
Step 7: Set each factor equal to zero and solve for x. This gives two possible solutions: 7x + 15 = 0 or x - 1 = 0.
For 7x + 15 = 0, subtracting 15 from both sides gives: 7x = -15. Dividing both sides by 7 yields: x = -15/7, which is one solution.
For x - 1 = 0, adding 1 to both sides gives: x = 1, which is another solution.
Step 8: Check for extraneous solutions. To do this, substitute each potential solution back into the original equation and verify if it is valid.
Substituting x = -15/7 into the original equation gives: (5/(-15/7)) + (8/(-15/7 + 3)) = 7. Simplifying gives: -7/3 + 56/12 = 7. After finding the common denominator and combining the fractions, it simplifies to: (-28 + 56)/12 = 7. Which then simplifies to: 28/12 = 7. Reducing the fraction gives: 7/3 = 7. Since this is not true, x = -15/7 is an extraneous solution.
Substituting x = 1 into the original equation gives: (5/1) + (8/(1 + 3)) = 7. Simplifying gives: 5 + 8/4 = 7. Converting 8/4 to the decimal form gives: 5 + 2 = 7. Since both sides of the equation are equal, x = 1 is a valid solution.
Therefore, the only solution to the equation is x = 1.
solve the rational equation indicate any extraneous solutions 4(x-3)/36-x^2 = 1/6-x+2x/6+x
To solve the rational equation (4(x-3))/(36-x^2) = (1)/(6-x) + (2x)/(6+x), we can follow these steps:
Step 1: Simplify both sides of the equation.
On the left side, we can simplify the expression by factoring the denominator. The denominator 36 - x^2 can be factored as (6 - x)(6 + x). Therefore, the equation becomes:
(4(x-3))/((6 - x)(6 + x)) = (1)/(6-x) + (2x)/(6+x).
Step 2: Find a common denominator.
The common denominator for the fractions on the right side is (6 - x)(6 + x). We can rewrite the equation with the common denominator:
(4(x-3))/((6 - x)(6 + x)) = ((1)(6+x))/((6-x)(6+x)) + ((2x)(6-x))/((6+x)(6-x)).
Step 3: Simplify and combine like terms.
The equation becomes:
(4(x-3))/((6 - x)(6 + x)) = (6+x + 12x - 2x^2)/((6+x)(6-x)).
Step 4: Multiply both sides of the equation by the common denominator.
By multiplying both sides by (6 - x)(6 + x), we can eliminate the denominators:
(4(x-3)) = (6+x + 12x - 2x^2).
Step 5: Simplify and bring all terms to one side.
Expand and simplify the right side:
4x - 12 = 18x - 2x^2.
Rearrange the equation:
2x^2 - 14x + 12 = 0.
Step 6: Factor the quadratic equation.
To factor the quadratic equation 2x^2 - 14x + 12 = 0, we can divide each term by 2 to simplify the equation:
x^2 - 7x + 6 = 0.
Factor the simplified equation:
(x - 1)(x - 6) = 0.
Step 7: Set each factor equal to zero and solve for x.
Set x - 1 = 0, which gives x = 1.
Set x - 6 = 0, which gives x = 6.
Step 8: Check for extraneous solutions.
We need to check if these solutions are valid or if they are extraneous by substituting them back into the original equation.
For x = 1, substituting into the original equation gives:
(4(1-3))/(36-1^2) = (1)/(6-1) + (2(1))/(6+1).
(-4)/35 = 1/5 + 2/7.
The equation is not true, so x = 1 is an extraneous solution.
For x = 6, substituting into the original equation gives:
(4(6-3))/(36-6^2) = (1)/(6-6) + (2(6))/(6+6).
(4)(3)/(36-36) = 1/12 + 12/12.
12/0 is undefined, so x = 6 is also an extraneous solution.
Therefore, there are no valid solutions to the equation.