Determine the Domain of the Fraction Equation: 5x/(2(x-7)) = 10/(8x)

Q: Why can't denominators equal zero?

Division by zero is undefined in mathematics! When a denominator equals zero, the fraction doesn't exist, so we must exclude those x-values from the domain.

Q: Do I need to solve the whole equation to find the domain?

No! The domain only depends on where denominators equal zero. You don't need to solve $ \frac{5x}{2(x-7)} = \frac{10}{8x} $ - just find where denominators are zero.

Q: What if a denominator has multiple terms like 2(x-7)?

Set the entire expression equal to zero: $ 2(x-7) = 0 $. Since 2 ≠ 0, we only need $ x-7 = 0 $, so $ x = 7 $.

Q: How do I write the final domain answer?

Use inequality notation: $ x ≠ 0, x ≠ 7 $ or interval notation: $ (-∞,0) ∪ (0,7) ∪ (7,∞) $. Both show x can be any real number except 0 and 7.

Q: What if I accidentally include a restricted value?

Your equation would be undefined! For example, if x = 0, then $ \frac{10}{8x} = \frac{10}{0} $ which doesn't exist. Always double-check your domain restrictions.

Domain Restrictions with Rational Equations

Find the domain

(no need to resolve)

$\frac{5x}{2(x-7)}=\frac{10}{8x}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 We're figuring out the range without solving the whole equation.

00:15 To find the range, let's make sure the denominator isn't zero.

00:19 Remember, dividing by zero isn't allowed.

00:23 First, we'll set the denominator equal to zero and check it out.

00:28 Let's simplify things as much as possible.

00:32 Next step, we isolate X.

00:37 Great! We've found the range from the first fraction.

00:41 Time to check the second denominator. Set that to zero too.

00:46 Again, let's focus on isolating X.

00:50 And there we have it. This is the second substitution range.

00:54 And that's how we solve the question. Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the domain

(no need to resolve)

$\frac{5x}{2(x-7)}=\frac{10}{8x}$

Step-by-step solution

To find the domain of the rational equation $\frac{5x}{2(x-7)} = \frac{10}{8x}$ , we need to ensure neither denominator equals zero.

Start by examining the first denominator, $2(x-7)$ :

Set $2(x-7) = 0$ .
Solve for $x$ to find $x - 7 = 0$ , resulting in $x = 7$ .

Next, examine the second denominator, $8x$ :

Set $8x = 0$ .
Solve for $x$ to find $x = 0$ .

Therefore, the function is undefined at $x = 0$ and $x = 7$ . These values should be excluded from the domain.

Thus, the domain of the given rational equation is all real numbers except where $x = 0$ and $x = 7$ .

This corresponds to the correct answer choice: $x \neq 0, x \neq 7$ .

Final Answer

$x≠0,x≠7$

Key Points to Remember

Essential concepts to master this topic

Rule: Domain excludes all values that make any denominator zero
Technique: Set each denominator equal to zero: 2(x-7) = 0 gives x = 7
Check: Verify both restrictions: x ≠ 0 and x ≠ 7 keep all denominators nonzero ✓

Common Mistakes

Avoid these frequent errors

Only checking one denominator for restrictions
Don't just find x ≠ 7 and ignore the second fraction = incomplete domain! This misses critical values where the equation is undefined. Always check every single denominator in the equation for zero values.

Practice Quiz

Test your knowledge with interactive questions

Select the the domain of the following fraction:

$\frac{6}{x}$

FAQ

Everything you need to know about this question

Why can't denominators equal zero?

Division by zero is undefined in mathematics! When a denominator equals zero, the fraction doesn't exist, so we must exclude those x-values from the domain.

Do I need to solve the whole equation to find the domain?

No! The domain only depends on where denominators equal zero. You don't need to solve $\frac{5x}{2(x-7)} = \frac{10}{8x}$ - just find where denominators are zero.

What if a denominator has multiple terms like 2(x-7)?

Set the entire expression equal to zero: $2(x-7) = 0$ . Since 2 ≠ 0, we only need $x-7 = 0$ , so $x = 7$ .

How do I write the final domain answer?

Use inequality notation: $x ≠ 0, x ≠ 7$ or interval notation: $(-∞,0) ∪ (0,7) ∪ (7,∞)$ . Both show x can be any real number except 0 and 7.

What if I accidentally include a restricted value?

Your equation would be undefined! For example, if x = 0, then $\frac{10}{8x} = \frac{10}{0}$ which doesn't exist. Always double-check your domain restrictions.

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Determine the Domain of the Fraction Equation: 5x/(2(x-7)) = 10/(8x)

Domain Restrictions 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 denominators equal zero?

Do I need to solve the whole equation to find the domain?

What if a denominator has multiple terms like 2(x-7)?

How do I write the final domain answer?

What if I accidentally include a restricted value?

🌟 Unlock Your Math Potential

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