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Solve each equation, if possible. $$ \frac{x}{x+2}=\frac{3}{2} $$

Short Answer

Expert verified
The solution is x = -6.

Step by step solution

01

Set up the equation

Given equation is \[\frac{x}{x+2} = \frac{3}{2}\]
02

Cross multiply

Cross-multiply to eliminate the fractions: \[x \times 2 = 3 \times (x+2)\]This gives us: \[2x = 3(x + 2)\]
03

Distribute and simplify

Distribute the 3 on the right side: \[2x = 3x + 6\]
04

Isolate x

Subtract 3x from both sides to isolate x: \[2x - 3x = 6\]This simplifies to: \[-x = 6\]
05

Solve for x

Multiply both sides by -1 to solve for x: \[x = -6\]
06

Verify the solution

Substitute \(x = -6\) back into the original equation to check:\[\frac{-6}{-6+2} = \frac{-6}{-4} = \frac{3}{2}\]Since both sides of the equation are equal, \(x = -6\) is a valid solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cross-Multiplication
Cross-multiplication is an essential technique used to solve equations involving fractions. In this exercise, we have the equation \(\frac{x}{x+2} = \frac{3}{2}\). To eliminate the fractions and make solving easier, we cross-multiply. This means we multiply the numerator of one fraction by the denominator of the other and set these products equal to each other.
For our equation, we get:
\[x \times 2 = 3 \times (x + 2)\]
After cross-multiplying, it’s easier to solve the resulting equation since it no longer contains fractions. This step transforms the problem into a straightforward algebraic equation.
Isolating Variables
After cross-multiplying, the next step is to isolate the variable we want to solve for. Let’s continue with our equation from the previous step:
\[2x = 3(x + 2)\]
We distribute the 3 on the right side:
\[2x = 3x + 6\]
Our goal is to get all the terms involving the variable on one side of the equation. We do this by subtracting 3x from both sides:
\[2x - 3x = 6\]
which simplifies to:
\[-x = 6\]
Finally, to solve for x, we multiply both sides by -1:
\[x = -6\]
This step-by-step approach ensures we isolate the variable efficiently, making it simpler to find the solution.
Verification of Solution
Simply solving for the variable is not enough. We need to verify our solution to make sure it's correct. We do this by substituting the found value back into the original equation and checking if both sides are equal.
For our problem, we determined that x = -6. We substitute this back into the original equation:
\[\frac{-6}{-6 + 2} = \frac{-6}{-4} \]
Simplifying the right-hand side, we get:
\[\frac{-6}{-4} = \frac{3}{2}\]
Since both sides of the equation are equal, we confirm that x = -6 is indeed a valid solution. This verification step is crucial because it confirms that the solution works within the original equation.
Rational Expressions
A rational expression is a fraction where the numerator and/or the denominator are polynomials. In this problem, \(\frac{x}{x+2}\) and \(\frac{3}{2}\) are both examples of rational expressions.
When solving rational equations, the first step often involves eliminating the denominators by using techniques like cross-multiplication.
One important aspect of working with rational expressions is to watch for any values that would make the denominator zero, as these are undefined. In our given equation, we need to ensure that x+2 ≠ 0, which happens if x ≠ -2. Fortunately, our solution x = -6 does not make the denominator zero.
Understanding rational expressions and their properties is key to solving and verifying equations that involve them, ensuring that solutions are valid and feasible.

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Most popular questions from this chapter

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