What value of satisfies the equation above?

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SAT Nonlinear Equations & Systems Question 66 | Free SAT Prep | Aniko

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Question 66·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

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\frac{6x^{2}}{x^{2}-16}-\frac{3x}{x-4}=\frac{2}{x+4}

What value of $x$ satisfies the equation above?

For rational equations on the SAT, start by factoring all denominators and marking any values that make a denominator zero, since these cannot be solutions. Next, either combine fractions using a common denominator or multiply through by the least common denominator to clear fractions. Simplify carefully, cancel common factors only when allowed (being mindful of domain restrictions), and solve the resulting simpler equation. Always check your final answer against the original denominators to rule out any values that make the expression undefined.

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Look at the denominators

Before doing any algebra, identify values of $x$ that would make the denominators $x^2 - 16$ , $x-4$ , or $x+4$ equal to zero. Those values can never be solutions.

Factor and find a common denominator

Notice that $x^2 - 16$ is a difference of squares. Factor it and then rewrite both fractions on the left-hand side using the same denominator.

Simplify before solving

After you combine the fractions on the left, look for common factors in the numerator and denominator that you can cancel (being careful about which $x$ -values are allowed). This should reduce the equation to something much simpler.

Clear the remaining denominator

Once both sides have the same denominator, think about how you can eliminate that denominator to get a simple equation in $x$ .

Desmos Guide

Enter each side of the equation as separate functions

Type f(x) = 6x^2/(x^2 - 16) - 3x/(x - 4) and g(x) = 2/(x + 4) into Desmos so you can see both graphs on the same coordinate plane.

Adjust the view to avoid the asymptotes

Zoom in around the origin so you can clearly see where the graphs of $f(x)$ and $g(x)$ might intersect, staying away from $x = 4$ and $x = -4$ , where there are vertical asymptotes.

Find the intersection point

Tap or click near the point where the graphs of $f(x)$ and $g(x)$ intersect; Desmos will display the coordinates of this point. The $x$ -coordinate of this intersection is the solution to the equation.

Step-by-step Explanation

Identify domain restrictions and factor the denominator

First, note that the denominators cannot be zero.

From $x^2 - 16$ , we get $x^2 - 16 = (x-4)(x+4)$ , so $x \neq 4$ and $x \neq -4$ .
From $x-4$ in the second term, we again get $x \neq 4$ .
From $x+4$ in the right-hand side, we again get $x \neq -4$ .

So any solution must satisfy $x \neq 4$ and $x \neq -4$ , and we can rewrite the equation as:

\frac{6x^{2}}{(x-4)(x+4)} - \frac{3x}{x-4} = \frac{2}{x+4}.

Combine the fractions on the left-hand side

To combine the two fractions on the left, use the common denominator $(x-4)(x+4)$ .

Rewrite the second term:

\frac{3x}{x-4} = \frac{3x(x+4)}{(x-4)(x+4)}.

Now subtract the numerators over the common denominator:

\frac{6x^{2}}{(x-4)(x+4)} - \frac{3x(x+4)}{(x-4)(x+4)} = \frac{6x^2 - 3x(x+4)}{(x-4)(x+4)}.

Simplify the numerator:

6x^2 - 3x(x+4) = 6x^2 - 3x^2 - 12x = 3x^2 - 12x = 3x(x-4).

So the left-hand side becomes:

\frac{3x(x-4)}{(x-4)(x+4)}.

Simplify the rational expression

Now cancel the common factor $(x-4)$ in the numerator and denominator, remembering that we already know $x \neq 4$ so this cancellation is allowed:

\frac{3x(x-4)}{(x-4)(x+4)} = \frac{3x}{x+4}.

So the original equation becomes the simpler equation:

\frac{3x}{x+4} = \frac{2}{x+4}, \quad x \neq -4.

Solve the simplified equation and check the solution

Since $x \neq -4$ , the factor $x+4$ in the denominator is not zero, so we can multiply both sides of

\frac{3x}{x+4} = \frac{2}{x+4}

by $x+4$ to clear the denominators:

3x = 2.

Solve for $x$ :

x = \frac{2}{3}.

Finally, check that $x = \frac{2}{3}$ does not make any denominator zero (it does not), so it is a valid solution. The value of $x$ that satisfies the equation is $\dfrac{2}{3}$ .

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Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation