Question

Solve each equation. $$ \frac{12}{3 x^{2}+12 x}=1-\frac{1}{x+4} $$

   Solve each equation.
$$
\frac{12}{3 x^{2}+12 x}=1-\frac{1}{x+4}
$$

Intermediate Algebra

Intermediate Algebra

Elayn Martin-Gay 6th Edition

Chapter 6, Problem 22

Instant Answer

Solved by Expert Charles Carter

07/17/2020

Step 1

Note 3x^2+12x = 3x(x+4), so the equation is 4/[x(x+4)] = 1 - 1/(x+4). Also x ≠ 0, -4 (denominators cannot be zero). Show more…

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Solve each equation. $$ \frac{12}{3 x^{2}+12 x}=1-\frac{1}{x+4} $$

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Key Concepts

Rational Equations

Rational equations involve variables in the denominator of one or more fractions. Solving them typically requires finding a common denominator to combine terms and eliminate the fractions, but with careful attention to restrictions on the variable that prevent division by zero.

Domain Restrictions

When working with rational expressions, it is critical to identify values for the variable that would make any denominator zero, as these values are not part of the solution set. This step ensures that all potential solutions are valid in the original equation.

Factoring

Factoring is the process of rewriting an algebraic expression as a product of simpler expressions. This technique is often used to simplify complex fractions and reveal common factors that can be canceled, thereby making the equation easier to solve.

Clearing Denominators

Clearing denominators involves multiplying every term in the equation by a common denominator to eliminate fractions and convert the problem into a polynomial equation. This method simplifies the equation but must be used carefully to avoid introducing extraneous solutions.

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