Solve the Fraction Equation: 9/(x+5) = 11/(2-x)

Q: Why can't I just multiply both sides by (x+5)(2-x)?

You absolutely can! That's actually the LCD method. Cross-multiplication is just a shortcut that gives the same result. Both methods work - cross-multiplication is often faster for simple rational equations.

Q: How do I know when to use cross-multiplication?

Use cross-multiplication when you have exactly one fraction on each side of the equation, like $ \frac{a}{b} = \frac{c}{d} $. For more complex equations with multiple terms, use the LCD method instead.

Q: Why is my answer a negative fraction?

Negative answers are completely normal! The fraction $ -\frac{37}{20} $ means x is between -2 and -1, which makes sense when you check the original equation.

Q: Should I convert the fraction to decimal form?

Keep it as a fraction! $ -\frac{37}{20} $ is the exact answer. Converting to -1.85 introduces rounding errors and is less precise than the fractional form.

Cross-Multiplication with Rational Equations

Solve for X:

$\frac{9}{x+5}=\frac{11}{2-x}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:04 Multiply by the common denominator to eliminate fractions

00:18 Make sure to open parentheses properly, multiply by each factor

00:34 Arrange the equation so that one side has only the unknown X

00:55 Collect like terms

01:09 Isolate X

01:19 Simplify as much as possible

01:24 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for X:

$\frac{9}{x+5}=\frac{11}{2-x}$

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Cross-multiply to eliminate the fractions.
Step 2: Solve the resulting linear equation.

Let's proceed step-by-step:

Step 1: Given the equation $\frac{9}{x+5} = \frac{11}{2-x}$ , we will cross-multiply:

$9 \times (2 - x) = 11 \times (x + 5)$

Simplify both sides:

$18 - 9x = 11x + 55$

Step 2: Solve for $x$ .

First, rearrange the terms to get all terms involving $x$ on one side:

$18 - 55 = 11x + 9x$

$-37 = 20x$

Divide both sides by 20 to solve for $x$ :

$x = -\frac{37}{20}$

Thus, the solution to the problem is $x = -\frac{37}{20}$ .

Final Answer

$-\frac{37}{20}$

Key Points to Remember

Essential concepts to master this topic

Cross-Multiplication Rule: Multiply diagonally across equal fractions to clear denominators
Technique: From $\frac{9}{x+5} = \frac{11}{2-x}$ get 9(2-x) = 11(x+5)
Verification: Substitute $x = -\frac{37}{20}$ back into original equation to confirm both sides equal ✓

Common Mistakes

Avoid these frequent errors

Incorrectly distributing negative signs during cross-multiplication
Don't write 9(2-x) = 18-9x but forget the negative! Missing the negative sign gives 18+9x instead of 18-9x, leading to wrong solutions like x = 37/20. Always carefully distribute each term, especially negatives.

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

Why can't I just multiply both sides by (x+5)(2-x)?

You absolutely can! That's actually the LCD method. Cross-multiplication is just a shortcut that gives the same result. Both methods work - cross-multiplication is often faster for simple rational equations.

What if one of the denominators becomes zero with my answer?

Great question! If substituting your answer makes any denominator zero, then it's not a valid solution. Always check that x ≠ -5 and x ≠ 2 for this problem.

How do I know when to use cross-multiplication?

Use cross-multiplication when you have exactly one fraction on each side of the equation, like $\frac{a}{b} = \frac{c}{d}$ . For more complex equations with multiple terms, use the LCD method instead.

Why is my answer a negative fraction?

Negative answers are completely normal! The fraction $-\frac{37}{20}$ means x is between -2 and -1, which makes sense when you check the original equation.

Should I convert the fraction to decimal form?

Keep it as a fraction! $-\frac{37}{20}$ is the exact answer. Converting to -1.85 introduces rounding errors and is less precise than the fractional form.

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Solve the Fraction Equation: 9/(x+5) = 11/(2-x)

Cross-Multiplication with Rational Equations

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I just multiply both sides by (x+5)(2-x)?

What if one of the denominators becomes zero with my answer?

How do I know when to use cross-multiplication?

Why is my answer a negative fraction?

Should I convert the fraction to decimal form?

🌟 Unlock Your Math Potential

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