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Math  /  Algebra

QuestionQuestion 19 of 42 Step 2 of 2 01:58:46
Consider the following equation: −2xx+1=−2+1x+3\frac{-2 x}{x+1}=-2+\frac{1}{x+3}
Step 2 of 2: Solve the equation, if possible. If there is a solution, express your answer as either an integer or a simplified fraction.
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Studdy Solution

STEP 1

What is this asking? We're asked to solve the equation −2xx+1=−2+1x+3\frac{-2x}{x+1} = -2 + \frac{1}{x+3} for xx, and give the answer as an integer or simplified fraction. Watch out! We need to be careful when we find our solutions, since we can't divide by zero.
So, if we get a solution where x=−1x = -1 or x=−3x = -3, we'll have to throw it out!

STEP 2

1. Rewrite the equation
2. Multiply both sides by the common denominator
3. Expand and simplify the equation
4. Solve for xx
5. Check for extraneous solutions

STEP 3

To make it easier to work with, let's rewrite the **constant term** on the right-hand side as a fraction with the same denominator as the other term on that side: −2=−2(x+3)x+3-2 = \frac{-2(x+3)}{x+3}.
So now our equation looks like this: −2xx+1=−2(x+3)x+3+1x+3 \frac{-2x}{x+1} = \frac{-2(x+3)}{x+3} + \frac{1}{x+3}

STEP 4

Since the fractions on the right-hand side now have the **same denominator**, we can add them together: −2xx+1=−2(x+3)+1x+3 \frac{-2x}{x+1} = \frac{-2(x+3) + 1}{x+3}

STEP 5

To get rid of those pesky fractions, we'll multiply both sides of the equation by the **common denominator**, which is (x+1)(x+3)(x+1)(x+3).
Remember, we're doing this to clear the fractions and make the equation easier to solve!

STEP 6

(x+1)(x+3)⋅−2xx+1=(x+1)(x+3)⋅−2(x+3)+1x+3 (x+1)(x+3) \cdot \frac{-2x}{x+1} = (x+1)(x+3) \cdot \frac{-2(x+3) + 1}{x+3}

STEP 7

Notice that (x+1)(x+1) and (x+3)(x+3) appear on both the top and bottom of each side.
We can divide to one to simplify: (x+3)(−2x)=(x+1)(−2(x+3)+1) (x+3)(-2x) = (x+1)(-2(x+3) + 1)

STEP 8

Now, let's expand both sides of the equation: −2x2−6x=(x+1)(−2x−6+1) -2x^2 - 6x = (x+1)(-2x - 6 + 1) −2x2−6x=(x+1)(−2x−5) -2x^2 - 6x = (x+1)(-2x - 5) −2x2−6x=−2x2−5x−2x−5 -2x^2 - 6x = -2x^2 - 5x - 2x - 5

STEP 9

Combine **like terms** on the right-hand side: −2x2−6x=−2x2−7x−5 -2x^2 - 6x = -2x^2 - 7x - 5

STEP 10

−2x2+2x2−6x=−2x2+2x2−7x−5 -2x^2 + 2x^2 - 6x = -2x^2 + 2x^2 - 7x - 5 −6x=−7x−5 -6x = -7x - 5

STEP 11

−6x+7x=−7x+7x−5 -6x + 7x = -7x + 7x - 5 x=−5 x = -5

STEP 12

x=−5x = -5

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