Solve and Apply: Finding the Field of Application for (x+y:3)/(2x+6)=4

Q: What does 'field of application' mean in math?

The field of application is another term for domain - all the values that x can take without making the expression undefined. It's where the equation 'works'!

Q: Why can't the denominator be zero?

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

Q: How do I write 'x cannot equal -3' properly?

Use the notation $ x \neq -3 $ which means 'x is not equal to -3'. You can also write it as x ≠ -3 or say 'all real numbers except x = -3'.

Q: What if there are multiple terms in the denominator?

Set the entire denominator equal to zero and solve. For example, if the denominator is $ x^2 - 4 $, solve $ x^2 - 4 = 0 $ to find all restricted values.

Q: Do I need to worry about the numerator being zero?

No! A numerator can be zero - that just makes the whole fraction equal to zero, which is perfectly fine. Only worry about the denominator being zero.

Domain Restrictions with Rational Expressions

$\frac{x+y:3}{2x+6}=4$

What is the field of application of the equation?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 Let's find the domain of the function.

00:16 Remember, dividing by zero is not okay in math.

00:20 Since our denominator has a variable, it can't be zero.

00:25 So, we need to make sure the main denominator is not zero.

00:30 Now, let's figure out what makes the variable X not equal to zero.

00:36 Once we solve that, we have our answer. And that's how we tackle this problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\frac{x+y:3}{2x+6}=4$

What is the field of application of the equation?

Step-by-step solution

To solve this problem, we'll follow these steps to find the domain:

Step 1: Recognize that the expression $\frac{x+y:3}{2x+6}=4$ involves a fraction. The denominator $2x + 6$ must not be zero, as division by zero is undefined.
Step 2: Set the denominator equal to zero and solve for $x$ to find the values that must be excluded: $2x + 6 = 0$ .
Step 3: Solve $2x + 6 = 0$ :

$2x + 6 = 0$
$2x = -6$
$x = -3$

Step 4: Conclude that the domain of the function excludes $x = -3$ , meaning $x \neq -3$ .

Thus, the domain of the given expression is all real numbers except $x = -3$ . This translates to:

$x\operatorname{\ne}-3$

Final Answer

$x\operatorname{\ne}-3$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Denominator cannot equal zero in rational expressions
Technique: Set 2x + 6 = 0, solve to get x = -3
Check: Verify x = -3 makes denominator zero: 2(-3) + 6 = 0 ✓

Common Mistakes

Avoid these frequent errors

Solving the entire equation instead of finding domain restrictions
Don't solve (x+y:3)/(2x+6) = 4 for x and y values = wrong focus! The question asks for field of application (domain), not solutions. Always identify what makes the denominator zero first.

Practice Quiz

Test your knowledge with interactive questions

Solve the equation

$$ 5x-15=30 $$

FAQ

Everything you need to know about this question

What does 'field of application' mean in math?

The field of application is another term for domain - all the values that x can take without making the expression undefined. It's where the equation 'works'!

Why can't the denominator be zero?

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

How do I write 'x cannot equal -3' properly?

Use the notation $x \neq -3$ which means 'x is not equal to -3'. You can also write it as x ≠ -3 or say 'all real numbers except x = -3'.

What if there are multiple terms in the denominator?

Set the entire denominator equal to zero and solve. For example, if the denominator is $x^2 - 4$ , solve $x^2 - 4 = 0$ to find all restricted values.

Do I need to worry about the numerator being zero?

No! A numerator can be zero - that just makes the whole fraction equal to zero, which is perfectly fine. Only worry about the denominator being zero.

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Solve and Apply: Finding the Field of Application for (x+y:3)/(2x+6)=4

Domain Restrictions with Rational Expressions

❤️ 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

What does 'field of application' mean in math?

Why can't the denominator be zero?

How do I write 'x cannot equal -3' properly?

What if there are multiple terms in the denominator?

Do I need to worry about the numerator being zero?

🌟 Unlock Your Math Potential

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