Verify the Rational Equation: Is (x²-64)/(x+8) = (x²+64)/(x+8) True?

Rational Equation Analysis with False Statements

Is the equation a true or false statement?

x2−64x+8=x2+64x+8 \frac{x^2-64}{x+8}=\frac{x^2+64}{x+8}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Are the expressions equal?
00:06 Let's multiply by denominators to reduce them
00:15 Let's reduce what we can
00:26 This equation is incorrect, therefore the expressions are not equal
00:29 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Is the equation a true or false statement?

x2−64x+8=x2+64x+8 \frac{x^2-64}{x+8}=\frac{x^2+64}{x+8}

2

Step-by-step solution

To solve the problem, we first recognize the potential problem in the equation due to the denominator:

  • The denominator x+8 x + 8 implies x≠−8 x \neq -8 . Otherwise, the expression is undefined.

Next, we will compare the numerators directly:

  • Numerator of the left-hand side: x2−64 x^2 - 64
  • Numerator of the right-hand side: x2+64 x^2 + 64

The expressions x2−64 x^2 - 64 and x2+64 x^2 + 64 are not equal for any real number x x , since adding 64 to one is not the same as subtracting 64 from the other. Therefore:

The equation is false for all x≠−8 x \neq -8 that are in the domain of the function, as clearly x2−64≠x2+64 x^2 - 64 \neq x^2 + 64 .

Thus, the statement is a Lie.

3

Final Answer

Lie

Key Points to Remember

Essential concepts to master this topic
  • Domain Restriction: Expressions undefined when denominator equals zero
  • Technique: Compare numerators directly: x2−64≠x2+64 x^2 - 64 \neq x^2 + 64
  • Check: Test specific values: when x=0, left side = -8, right side = 8 ✓

Common Mistakes

Avoid these frequent errors
  • Canceling denominators without checking if numerators are equal
    Don't assume equal denominators mean equal fractions = wrong conclusion! The numerators x2−64 x^2 - 64 and x2+64 x^2 + 64 are never equal. Always compare numerators when denominators are identical.

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 we just cancel out the denominators since they're the same?

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You can only cancel denominators when the entire fractions are equal! Here, even though both sides have x+8 x + 8 in the denominator, the numerators are different, so the fractions aren't equal.

What does it mean when x ≠ -8?

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This is called a domain restriction. When x=−8 x = -8 , the denominator becomes zero, making the expression undefined. We must exclude this value from our consideration.

How can I tell if a rational equation is true or false?

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Compare the simplified forms of both sides. If they're identical for all valid x-values, it's true. If they differ for any valid x-value, it's false.

Could this equation ever be true for some specific x-value?

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No! For any real number x, subtracting 64 gives a different result than adding 64. The difference (x2+64)−(x2−64)=128 (x^2 + 64) - (x^2 - 64) = 128 is always 128, never zero.

What's the difference between 'Truth' and 'True' in the answer choices?

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Both Truth and True would mean the equation is correct. The key insight is recognizing that this equation is actually false (a lie) for all valid x-values.

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