Solve for X: Complex Fraction Equation (8-3(x-2))/(5-x) = 4/3

Q: Why can't I just cross multiply right away?

You can! Since we have $ \frac{8-3(x-2)}{5-x} = \frac{4}{3} $, cross multiplying gives: 3[8-3(x-2)] = 4(5-x). This is often faster than clearing denominators step by step.

Q: What if 5-x equals zero in the denominator?

Great catch! If x = 5, the denominator becomes zero and the fraction is undefined. Always check that your solution doesn't make any denominator equal to zero.

Q: How do I handle the distribution of -3(x-2)?

Remember: negative times negative equals positive. So -3(x-2) = -3×x + (-3)×(-2) = -3x + 6. Don't forget that second term becomes positive!

Q: Should I convert 22/5 to a decimal?

Keep it as a fraction! $ \frac{22}{5} $ is the exact answer. Converting to 4.4 introduces rounding and makes verification harder.

Rational Equations with Cross Multiplication

Solve for X:

$\frac{8-3(x-2)}{5-x}=\frac{4}{3}$

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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 Make sure to open parentheses properly, multiply by each factor

01:03 Collect like terms

01:07 Arrange the equation so that X is isolated on one side

01:26 Collect like terms

01:32 Isolate X

01:38 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:

$\frac{8-3(x-2)}{5-x}=\frac{4}{3}$

Step-by-step solution

We will solve the equation $\frac{8-3(x-2)}{5-x} = \frac{4}{3}$ step by step.

First, clear the fraction by multiplying both sides of the equation by $5-x$ :

8 - 3(x-2) = \frac{4}{3} \cdot (5-x)

Distribute the $-3$ on the left side:

8 - 3x + 6 = \frac{4}{3}(5-x)

Combine like terms on the left side:

14 - 3x = \frac{4}{3}(5-x)

Now, clear the fraction on the right side by multiplying through by 3:

3(14 - 3x) = 4(5-x)

Distribute the values on both sides:

42 - 9x = 20 - 4x

Rearrange the equation to isolate terms with $x$ :

42 - 20 = 9x - 4x

Simplify the equation:

22 = 5x

Solve for $x$ by dividing both sides by 5:

x = \frac{22}{5}

Therefore, the solution to the problem is $x = \frac{22}{5}$ .

Final Answer

$\frac{22}{5}$

Key Points to Remember

Essential concepts to master this topic

Rule: Cross multiply or clear denominators to eliminate fractions completely
Technique: Multiply both sides by (5-x): 8-3(x-2) = 4(5-x)/3
Check: Substitute x = 22/5 back into original equation ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute negative signs when expanding
Don't expand -3(x-2) as -3x-6 = wrong signs everywhere! The negative distributes to both terms. Always multiply -3 × x = -3x AND -3 × (-2) = +6, giving -3x + 6.

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 cross multiply right away?

You can! Since we have $\frac{8-3(x-2)}{5-x} = \frac{4}{3}$ , cross multiplying gives: 3[8-3(x-2)] = 4(5-x). This is often faster than clearing denominators step by step.

What if 5-x equals zero in the denominator?

Great catch! If x = 5, the denominator becomes zero and the fraction is undefined. Always check that your solution doesn't make any denominator equal to zero.

How do I handle the distribution of -3(x-2)?

Remember: negative times negative equals positive. So -3(x-2) = -3×x + (-3)×(-2) = -3x + 6. Don't forget that second term becomes positive!

Should I convert 22/5 to a decimal?

Keep it as a fraction! $\frac{22}{5}$ is the exact answer. Converting to 4.4 introduces rounding and makes verification harder.

How can I check my work without substituting back?

While substitution is best, you can also work backwards: start with x = 22/5 and see if you get the original equation. But honestly, substitution is quicker and more reliable!

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Solve for X: Complex Fraction Equation (8-3(x-2))/(5-x) = 4/3

Rational Equations with Cross Multiplication

❤️ 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 cross multiply right away?

What if 5-x equals zero in the denominator?

How do I handle the distribution of -3(x-2)?

Should I convert 22/5 to a decimal?

How can I check my work without substituting back?

🌟 Unlock Your Math Potential

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