Solve for X: Finding the Value in 9/x = 3/(x+2)

Rational Equations with Cross-Multiplication

Solve for X:

9x=3x+2 \frac{9}{x}=\frac{3}{x+2}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find X
00:05 Multiply by denominators to eliminate fractions
00:26 Simplify as much as possible
00:35 Carefully open parentheses properly, multiply by each term
00:48 Arrange the equation so that only the unknown X is on one side
01:01 Combine like terms
01:08 Isolate X
01:15 Simplify as much as possible
01:20 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Solve for X:

9x=3x+2 \frac{9}{x}=\frac{3}{x+2}

2

Step-by-step solution

To solve the equation 9x=3x+2 \frac{9}{x} = \frac{3}{x+2} , we'll follow these steps:

  • Step 1: Use cross-multiplication to eliminate the fractions.

  • Step 2: Simplify the resulting linear equation.

  • Step 3: Solve for x x .

Now, let's work through each step:

Step 1: Cross-multiply to clear the fractions:

9x=3x+2\frac{9}{x} = \frac{3}{x+2}

Cross-multiplying gives:

9(x+2)=3x9(x + 2) = 3x

Step 2: Distribute the 9 on the left side:

9x+18=3x9x + 18 = 3x

Step 3: Isolate the variable x x :

Subtract 3x 3x from both sides:

9x+183x=3x3x9x + 18 - 3x = 3x - 3x

This simplifies to:

6x+18=06x + 18 = 0

Subtract 18 from both sides:

6x=186x = -18

Divide by 6 to solve for x x :

x=186x = \frac{-18}{6}

Therefore, x=3 x = -3 .

The solution to the problem is x=3 x = -3 .

3

Final Answer

3 -3

Key Points to Remember

Essential concepts to master this topic
  • Cross-Multiplication Rule: Multiply diagonally across equal fractions to eliminate denominators
  • Technique: 9x=3x+2 \frac{9}{x} = \frac{3}{x+2} becomes 9(x+2)=3x 9(x+2) = 3x
  • Check: Substitute x = -3: 93=3 \frac{9}{-3} = -3 and 31=3 \frac{3}{-1} = -3

Common Mistakes

Avoid these frequent errors
  • Forgetting to distribute after cross-multiplication
    Don't write 9(x+2) = 3x and then just cancel terms = wrong answer! Students often skip the distribution step and get confused. Always distribute first: 9x + 18 = 3x, then solve systematically.

Practice Quiz

Test your knowledge with interactive questions

Solve for X:

\( x - 3 + 5 = 8 - 2 \)

FAQ

Everything you need to know about this question

When can I use cross-multiplication?

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Cross-multiplication works when you have one fraction equals another fraction, like ab=cd \frac{a}{b} = \frac{c}{d} . It's perfect for rational equations but won't work if you have addition or subtraction between fractions.

What if I get a negative answer like x = -3?

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Negative solutions are completely normal in algebra! Just make sure to check your work by substituting back. As long as the denominators aren't zero, negative answers are valid.

How do I know if my denominator will be zero?

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Always check that your answer doesn't make any denominator equal zero. In this problem, x = -3 makes the first denominator -3 (not zero) and the second denominator -1 (not zero), so it's valid!

Why do I distribute 9(x+2) instead of just canceling?

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Distribution is required because you have multiplication across parentheses. You can't cancel or skip this step - it's like saying 9 × (5+3) equals 9 × 5, which isn't true!

Can I solve this equation a different way?

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Yes! You could multiply both sides by x(x+2) x(x+2) to clear fractions, but cross-multiplication is usually faster and cleaner for equations with two fractions.

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