Question

The equations in Exercises 79–90 combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 4x/(x + 3) - 12/(x - 3) = (4x2 + 36)/(x2 - 9)

Accepted Answer

Rewrite the equation to identify the denominators. Notice that the denominators are $$x + 3$$, $$x - 3$$, and $$x^2 - 9. $$Recognize that $$x^2 - 9 is a $$difference of squares and can be factored as $$(x + 3)(x - 3). $$Multiply through the entire equation by the least common denominator (LCD), which is $$(x + 3)(x - 3)$$, to eliminate the fractions. This will simplify the equation significantly. Distribute the LCD to each term in the equation. For the first term, $$\frac{4x}{x + 3}$$, multiplying by $$(x + 3)(x - 3)$$ leaves $$4x(x - 3). $$For the second term, $$\frac{-12}{x - 3}$$, multiplying by $$(x + 3)(x - 3)$$ leaves $$-12(x + 3). $$For the right-hand side, $$\frac{4x^2 + 36}{x^2 - 9}$$, multiplying by $$(x + 3)(x - 3)$$ leaves $$4x^2 + 36. $$Simplify the resulting equation by distributing and combining like terms. Expand $$4x(x - 3)$$ and $$-12(x + 3)$$, then combine all terms on one side of the equation to form a standard quadratic equation. Solve the quadratic equation using factoring, the quadratic formula, or completing the square. After finding the solutions, check for any restrictions on $$x (e.g$$., $$x 
eq -3$$ and $$x 
eq 3$$ because these values make the original denominators undefined). Determine whether the equation is an identity (true for all values of $$x$$), a conditional equation (true for specific $$x$$ values), or inconsistent (no solution).

The equations in Exercises 79–90 combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 4x/(x + 3) - 12/(x - 3) = (4x² + 36)/(x² - 9)

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Types of Equations

Rational Expressions

Factoring Polynomials