Algebra Examples

$\frac{8}{(x - 7) (x + 3)} = \frac{7}{3 x - 10}$

Step 1

Subtract $\frac{7}{3 x - 10}$ from both sides of the equation.

$\frac{8}{(x - 7) (x + 3)} - \frac{7}{3 x - 10} = 0$

Step 2

Simplify $\frac{8}{(x - 7) (x + 3)} - \frac{7}{3 x - 10}$ .

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Step 2.1

To write $\frac{8}{(x - 7) (x + 3)}$ as a fraction with a common denominator, multiply by $\frac{3 x - 10}{3 x - 10}$ .

$\frac{8}{(x - 7) (x + 3)} \cdot \frac{3 x - 10}{3 x - 10} - \frac{7}{3 x - 10} = 0$

Step 2.2

To write $- \frac{7}{3 x - 10}$ as a fraction with a common denominator, multiply by $\frac{(x - 7) (x + 3)}{(x - 7) (x + 3)}$ .

$\frac{8}{(x - 7) (x + 3)} \cdot \frac{3 x - 10}{3 x - 10} - \frac{7}{3 x - 10} \cdot \frac{(x - 7) (x + 3)}{(x - 7) (x + 3)} = 0$

Step 2.3

Write each expression with a common denominator of $(x - 7) (x + 3) (3 x - 10)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{8}{(x - 7) (x + 3)}$ by $\frac{3 x - 10}{3 x - 10}$ .

$\frac{8 (3 x - 10)}{(x - 7) (x + 3) (3 x - 10)} - \frac{7}{3 x - 10} \cdot \frac{(x - 7) (x + 3)}{(x - 7) (x + 3)} = 0$

Step 2.3.2

Multiply $\frac{7}{3 x - 10}$ by $\frac{(x - 7) (x + 3)}{(x - 7) (x + 3)}$ .

$\frac{8 (3 x - 10)}{(x - 7) (x + 3) (3 x - 10)} - \frac{7 ((x - 7) (x + 3))}{(3 x - 10) ((x - 7) (x + 3))} = 0$

Step 2.3.3

Reorder the factors of $(x - 7) (x + 3) (3 x - 10)$ .

$\frac{8 (3 x - 10)}{(3 x - 10) (x + 3) (x - 7)} - \frac{7 ((x - 7) (x + 3))}{(3 x - 10) ((x - 7) (x + 3))} = 0$

Step 2.3.4

Reorder the factors of $(3 x - 10) ((x - 7) (x + 3))$ .

$\frac{8 (3 x - 10)}{(3 x - 10) (x + 3) (x - 7)} - \frac{7 ((x - 7) (x + 3))}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{8 (3 x - 10) - 7 ((x - 7) (x + 3))}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{8 (3 x) + 8 \cdot - 10 - 7 (x - 7) (x + 3)}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.2

Multiply $3$ by $8$ .

$\frac{24 x + 8 \cdot - 10 - 7 (x - 7) (x + 3)}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.3

Multiply $8$ by $- 10$ .

$\frac{24 x - 80 - 7 (x - 7) (x + 3)}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.4

Apply the distributive property.

$\frac{24 x - 80 + (- 7 x - 7 \cdot - 7) (x + 3)}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.5

Multiply $- 7$ by $- 7$ .

$\frac{24 x - 80 + (- 7 x + 49) (x + 3)}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.6

Expand $(- 7 x + 49) (x + 3)$ using the FOIL Method.

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Step 2.5.6.1

Apply the distributive property.

$\frac{24 x - 80 - 7 x (x + 3) + 49 (x + 3)}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.6.2

Apply the distributive property.

$\frac{24 x - 80 - 7 x \cdot x - 7 x \cdot 3 + 49 (x + 3)}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.6.3

Apply the distributive property.

$\frac{24 x - 80 - 7 x \cdot x - 7 x \cdot 3 + 49 x + 49 \cdot 3}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.7

Simplify and combine like terms.

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Step 2.5.7.1

Simplify each term.

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Step 2.5.7.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.5.7.1.1.1

Move $x$ .

$\frac{24 x - 80 - 7 (x \cdot x) - 7 x \cdot 3 + 49 x + 49 \cdot 3}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.7.1.1.2

Multiply $x$ by $x$ .

$\frac{24 x - 80 - 7 x^{2} - 7 x \cdot 3 + 49 x + 49 \cdot 3}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.7.1.2

Multiply $3$ by $- 7$ .

$\frac{24 x - 80 - 7 x^{2} - 21 x + 49 x + 49 \cdot 3}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.7.1.3

Multiply $49$ by $3$ .

$\frac{24 x - 80 - 7 x^{2} - 21 x + 49 x + 147}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.7.2

Add $- 21 x$ and $49 x$ .

$\frac{24 x - 80 - 7 x^{2} + 28 x + 147}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.8

Add $24 x$ and $28 x$ .

$\frac{52 x - 80 - 7 x^{2} + 147}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.9

Add $- 80$ and $147$ .

$\frac{52 x - 7 x^{2} + 67}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.5.10

Reorder terms.

$\frac{- 7 x^{2} + 52 x + 67}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.6

Factor $- 1$ out of $- 7 x^{2}$ .

$\frac{- (7 x^{2}) + 52 x + 67}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.7

Factor $- 1$ out of $52 x$ .

$\frac{- (7 x^{2}) - (- 52 x) + 67}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.8

Factor $- 1$ out of $- (7 x^{2}) - (- 52 x)$ .

$\frac{- (7 x^{2} - 52 x) + 67}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.9

Rewrite $67$ as $- 1 (- 67)$ .

$\frac{- (7 x^{2} - 52 x) - 1 (- 67)}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.10

Factor $- 1$ out of $- (7 x^{2} - 52 x) - 1 (- 67)$ .

$\frac{- (7 x^{2} - 52 x - 67)}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.11

Rewrite $- (7 x^{2} - 52 x - 67)$ as $- 1 (7 x^{2} - 52 x - 67)$ .

$\frac{- 1 (7 x^{2} - 52 x - 67)}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 2.12

Move the negative in front of the fraction.

$- \frac{7 x^{2} - 52 x - 67}{(3 x - 10) (x + 3) (x - 7)} = 0$

Step 3

Set the denominator in $\frac{7 x^{2} - 52 x - 67}{(3 x - 10) (x + 3) (x - 7)}$ equal to $0$ to find where the expression is undefined.

$(3 x - 10) (x + 3) (x - 7) = 0$

Step 4

Solve for $x$ .

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Step 4.1

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

$3 x - 10 = 0$

$x + 3 = 0$

$x - 7 = 0$

Step 4.2

Set $3 x - 10$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $3 x - 10$ equal to $0$ .

$3 x - 10 = 0$

Step 4.2.2

Solve $3 x - 10 = 0$ for $x$ .

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Step 4.2.2.1

Add $10$ to both sides of the equation.

$3 x = 10$

Step 4.2.2.2

Divide each term in $3 x = 10$ by $3$ and simplify.

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Step 4.2.2.2.1

Divide each term in $3 x = 10$ by $3$ .

$\frac{3 x}{3} = \frac{10}{3}$

Step 4.2.2.2.2

Simplify the left side.

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Step 4.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{10}{3}$

Step 4.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{3}$

Step 4.3

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 4.3.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4.4

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 4.4.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 4.4.2

Add $7$ to both sides of the equation.

$x = 7$

Step 4.5

The final solution is all the values that make $(3 x - 10) (x + 3) (x - 7) = 0$ true.

$x = \frac{10}{3}, - 3, 7$

Step 5

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 $0$ .

$x = - 3, x = \frac{10}{3}, x = 7$

Step 6