Solve the Fraction Equation: Find X When -7/(x+4) = x-4

Q: Why do I multiply both sides by (x+4)?

Multiplying by (x+4) eliminates the fraction on the left side! This transforms $ \frac{-7}{x+4} $ into just -7, making the equation much easier to solve.

Q: What's a difference of squares and why does it matter here?

A difference of squares follows the pattern $ a^2 - b^2 = (a+b)(a-b) $. Here, (x-4)(x+4) = x²-16, which helps us expand the right side quickly.

Q: How do I know if x = -4 is allowed as a solution?

Look at the original equation! Since we have $ \frac{-7}{x+4} $, setting x = -4 would make the denominator zero, which is undefined. So x = -4 is never allowed.

Q: Why do we get two solutions from this equation?

After clearing fractions, we get a quadratic equation $ x^2 - 9 = 0 $. Quadratic equations typically have two solutions, and both x = 3 and x = -3 are valid since neither makes the denominator zero.

Q: How can I check if my answers are correct?

Substitute each solution back into the original equation:For x = 3: $ \frac{-7}{3+4} = \frac{-7}{7} = -1 $ and x - 4 = 3 - 4 = -1 ✓For x = -3: $ \frac{-7}{-3+4} = \frac{-7}{1} = -7 $ and x - 4 = -3 - 4 = -7 ✓

Rational Equations with Quadratic Solutions

Resolve:

$\frac{-7}{x+4}=x-4$

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Understand the problem

Resolve:

$\frac{-7}{x+4}=x-4$

Step-by-step solution

To solve this equation, we follow these steps:

Step 1: Identify the original equation as $\frac{-7}{x+4} = x-4$ .
Step 2: Eliminate the fraction by multiplying both sides by $x+4$ :
$-7 = (x-4)(x+4)$ .
Step 3: Recognize the right-hand side as a difference of squares:
$-7 = x^2 - 16$ .
Step 4: Rearrange the equation into a standard quadratic form:
$x^2 - 16 + 7 = 0$ which simplifies to $x^2 - 9 = 0$ .
Step 5: Factor the quadratic expression:
$(x-3)(x+3) = 0$ .
Step 6: Use the zero product property to find potential solutions:
$x - 3 = 0$ which gives $x = 3$ , and
$x + 3 = 0$ which gives $x = -3$ .
Step 7: Validate that neither solution makes the denominator zero, as $x \neq -4$ .

Thus, the solutions to the equation $\frac{-7}{x+4} = x-4$ are $\pm3$ .

Final Answer

$\pm3$

Key Points to Remember

Essential concepts to master this topic

Clear Fractions: Multiply both sides by (x+4) to eliminate denominators
Expand: Use difference of squares: (x-4)(x+4) = x²-16
Verify: Check x≠-4 and substitute: -7/7 = -1 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check domain restrictions
Don't solve x²-9=0 and stop at x=±3 without checking restrictions = invalid solutions! The original equation has x+4 in denominator, so x≠-4. Always verify solutions don't make any denominator zero.

Practice Quiz

Test your knowledge with interactive questions

Solve:

$$ (2+x)(2-x)=0 $$

FAQ

Everything you need to know about this question

Why do I multiply both sides by (x+4)?

Multiplying by (x+4) eliminates the fraction on the left side! This transforms $\frac{-7}{x+4}$ into just -7, making the equation much easier to solve.

What's a difference of squares and why does it matter here?

A difference of squares follows the pattern $a^2 - b^2 = (a+b)(a-b)$ . Here, (x-4)(x+4) = x²-16, which helps us expand the right side quickly.

How do I know if x = -4 is allowed as a solution?

Look at the original equation! Since we have $\frac{-7}{x+4}$ , setting x = -4 would make the denominator zero, which is undefined. So x = -4 is never allowed.

Why do we get two solutions from this equation?

After clearing fractions, we get a quadratic equation $x^2 - 9 = 0$ . Quadratic equations typically have two solutions, and both x = 3 and x = -3 are valid since neither makes the denominator zero.

How can I check if my answers are correct?

Substitute each solution back into the original equation:

For x = 3: $\frac{-7}{3+4} = \frac{-7}{7} = -1$ and x - 4 = 3 - 4 = -1 ✓
For x = -3: $\frac{-7}{-3+4} = \frac{-7}{1} = -7$ and x - 4 = -3 - 4 = -7 ✓

What if I forgot to multiply the right side by (x+4)?

This is a common error! If you only clear the fraction on the left, you'll get -7 = x - 4, giving x = -3 as the only solution. You'd miss x = 3 completely! Always multiply both sides by the same expression.

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