Find Where Undefined/Discontinuous 1/(x-4)-5/(x+6)=10/(x^2+2x-24)

$\frac{1}{x - 4} - \frac{5}{x + 6} = \frac{10}{x^{2} + 2 x - 24}$

Move $\frac{10}{x^{2} + 2 x - 24}$ to the left side of the equation by subtracting it from both sides.

$\frac{1}{x - 4} - \frac{5}{x + 6} - \frac{10}{x^{2} + 2 x - 24} = 0$

Set the denominator in $\frac{1}{x - 4}$ equal to $0$ to find where the expression is undefined.

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Set the denominator in $\frac{5}{x + 6}$ equal to $0$ to find where the expression is undefined.

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

Set the denominator in $\frac{10}{x^{2} + 2 x - 24}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 2 x - 24 = 0$

Solve for $x$ .

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Factor $x^{2} + 2 x - 24$ using the AC method.

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Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $2$ .

$- 4, 6$

Write the factored form using these integers.

$(x - 4) (x + 6) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 4 = 0$

$x + 6 = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $(x - 4) (x + 6) = 0$ true.

$x = 4, - 6$

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 6, x = 4$

Find Where Undefined/Discontinuous 1/(x-4)-5/(x+6)=10/(x^2+2x-24)

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