Precalculus Examples

\frac{x - 2}{x + 1} + \frac{2 x - 7}{x - 6} = \frac{x - 3}{x^{2} - 5 x - 6}

$\frac{x - 2}{x + 1} + \frac{2 x - 7}{x - 6} = \frac{x - 3}{x^{2} - 5 x - 6}$

Step 1

Factor

$x^{2} - 5 x - 6$ using the AC method.

Tap for more steps...

Step 1.1

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

$- 6$ and whose sum is

$- 5$ .

$- 6, 1$

Step 1.2

Write the factored form using these integers.

$\frac{x - 2}{x + 1} + \frac{2 x - 7}{x - 6} = \frac{x - 3}{(x - 6) (x + 1)}$

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x + 1, x - 6, (x - 6) (x + 1)$

Step 2.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.3

The number

$1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.4

The LCM of

$1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 2.5

The factor for

$x + 1$ is

$x + 1$ itself.

$(x + 1) = x + 1$

$(x + 1)$ occurs

$1$ time.

Step 2.6

The factor for

$x - 6$ is

$x - 6$ itself.

$(x - 6) = x - 6$

$(x - 6)$ occurs

$1$ time.

Step 2.7

The factor for

$x + 1$ is

$x + 1$ itself.

$(x + 1) = x + 1$

$(x + 1)$ occurs

$1$ time.

Step 2.8

The LCM of

$x + 1, x - 6, x - 6, x + 1$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x + 1) (x - 6)$

Step 3

Multiply each term in

$\frac{x - 2}{x + 1} + \frac{2 x - 7}{x - 6} = \frac{x - 3}{(x - 6) (x + 1)}$ by

$(x + 1) (x - 6)$ to eliminate the fractions.

Tap for more steps...

Step 3.1

Multiply each term in

$\frac{x - 2}{x + 1} + \frac{2 x - 7}{x - 6} = \frac{x - 3}{(x - 6) (x + 1)}$ by

$(x + 1) (x - 6)$ .

$\frac{x - 2}{x + 1} ((x + 1) (x - 6)) + \frac{2 x - 7}{x - 6} ((x + 1) (x - 6)) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Cancel the common factor of

$x + 1$ .

Tap for more steps...

Step 3.2.1.1.1

Cancel the common factor.

$\frac{x - 2}{x + 1} ((x + 1) (x - 6)) + \frac{2 x - 7}{x - 6} ((x + 1) (x - 6)) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.1.2

Rewrite the expression.

$(x - 2) (x - 6) + \frac{2 x - 7}{x - 6} ((x + 1) (x - 6)) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.2

Expand

$(x - 2) (x - 6)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1.2.1

Apply the distributive property.

$x (x - 6) - 2 (x - 6) + \frac{2 x - 7}{x - 6} ((x + 1) (x - 6)) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 6 - 2 (x - 6) + \frac{2 x - 7}{x - 6} ((x + 1) (x - 6)) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 6 - 2 x - 2 \cdot - 6 + \frac{2 x - 7}{x - 6} ((x + 1) (x - 6)) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 3.2.1.3.1

Simplify each term.

Tap for more steps...

Step 3.2.1.3.1.1

Multiply

$x$ by

$x$ .

$x^{2} + x \cdot - 6 - 2 x - 2 \cdot - 6 + \frac{2 x - 7}{x - 6} ((x + 1) (x - 6)) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.3.1.2

Move

$- 6$ to the left of

$x$ .

$x^{2} - 6 \cdot x - 2 x - 2 \cdot - 6 + \frac{2 x - 7}{x - 6} ((x + 1) (x - 6)) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.3.1.3

Multiply

$- 2$ by

$- 6$ .

$x^{2} - 6 x - 2 x + 12 + \frac{2 x - 7}{x - 6} ((x + 1) (x - 6)) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.3.2

Subtract

$2 x$ from

$- 6 x$ .

$x^{2} - 8 x + 12 + \frac{2 x - 7}{x - 6} ((x + 1) (x - 6)) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.4

Cancel the common factor of

$x - 6$ .

Tap for more steps...

Step 3.2.1.4.1

Factor

$x - 6$ out of

$(x + 1) (x - 6)$ .

$x^{2} - 8 x + 12 + \frac{2 x - 7}{x - 6} ((x - 6) (x + 1)) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.4.2

Cancel the common factor.

$x^{2} - 8 x + 12 + \frac{2 x - 7}{x - 6} ((x - 6) (x + 1)) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.4.3

Rewrite the expression.

$x^{2} - 8 x + 12 + (2 x - 7) (x + 1) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.5

Expand

$(2 x - 7) (x + 1)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1.5.1

Apply the distributive property.

$x^{2} - 8 x + 12 + 2 x (x + 1) - 7 (x + 1) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.5.2

Apply the distributive property.

$x^{2} - 8 x + 12 + 2 x \cdot x + 2 x \cdot 1 - 7 (x + 1) = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.5.3

Apply the distributive property.

$x^{2} - 8 x + 12 + 2 x \cdot x + 2 x \cdot 1 - 7 x - 7 \cdot 1 = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.6

Simplify and combine like terms.

Tap for more steps...

Step 3.2.1.6.1

Simplify each term.

Tap for more steps...

Step 3.2.1.6.1.1

Multiply

$x$ by

$x$ by adding the exponents.

Tap for more steps...

Step 3.2.1.6.1.1.1

Move

$x$ .

$x^{2} - 8 x + 12 + 2 (x \cdot x) + 2 x \cdot 1 - 7 x - 7 \cdot 1 = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.6.1.1.2

Multiply

$x$ by

$x$ .

$x^{2} - 8 x + 12 + 2 x^{2} + 2 x \cdot 1 - 7 x - 7 \cdot 1 = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.6.1.2

Multiply

$2$ by

$1$ .

$x^{2} - 8 x + 12 + 2 x^{2} + 2 x - 7 x - 7 \cdot 1 = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.6.1.3

Multiply

$- 7$ by

$1$ .

$x^{2} - 8 x + 12 + 2 x^{2} + 2 x - 7 x - 7 = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.1.6.2

Subtract

$7 x$ from

$2 x$ .

$x^{2} - 8 x + 12 + 2 x^{2} - 5 x - 7 = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.2

Simplify by adding terms.

Tap for more steps...

Step 3.2.2.1

Add

$x^{2}$ and

$2 x^{2}$ .

$3 x^{2} - 8 x + 12 - 5 x - 7 = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.2.2

Subtract

$5 x$ from

$- 8 x$ .

$3 x^{2} - 13 x + 12 - 7 = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.2.2.3

Subtract

$7$ from

$12$ .

$3 x^{2} - 13 x + 5 = \frac{x - 3}{(x - 6) (x + 1)} ((x + 1) (x - 6))$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Cancel the common factor of

$(x + 1) (x - 6)$ .

Tap for more steps...

Step 3.3.1.1

Factor

$(x + 1) (x - 6)$ out of

$(x - 6) (x + 1)$ .

$3 x^{2} - 13 x + 5 = \frac{x - 3}{(x + 1) (x - 6) (1)} ((x + 1) (x - 6))$

Step 3.3.1.2

Cancel the common factor.

$3 x^{2} - 13 x + 5 = \frac{x - 3}{(x + 1) (x - 6) \cdot 1} ((x + 1) (x - 6))$

Step 3.3.1.3

Rewrite the expression.

$3 x^{2} - 13 x + 5 = x - 3$

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

Move all terms containing

$x$ to the left side of the equation.

Tap for more steps...

Step 4.1.1

Subtract

$x$ from both sides of the equation.

$3 x^{2} - 13 x + 5 - x = - 3$

Step 4.1.2

Subtract

$x$ from

$- 13 x$ .

$3 x^{2} - 14 x + 5 = - 3$

Step 4.2

Add

$3$ to both sides of the equation.

$3 x^{2} - 14 x + 5 + 3 = 0$

Step 4.3

Add

$5$ and

$3$ .

$3 x^{2} - 14 x + 8 = 0$

Step 4.4

Factor by grouping.

Tap for more steps...

Step 4.4.1

For a polynomial of the form

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

$a \cdot c = 3 \cdot 8 = 24$ and whose sum is

$b = - 14$ .

Tap for more steps...

Step 4.4.1.1

Factor

$- 14$ out of

$- 14 x$ .

$3 x^{2} - 14 x + 8 = 0$

Step 4.4.1.2

Rewrite

$- 14$ as

$- 2$ plus

$- 12$

$3 x^{2} + (- 2 - 12) x + 8 = 0$

Step 4.4.1.3

Apply the distributive property.

$3 x^{2} - 2 x - 12 x + 8 = 0$

Step 4.4.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.4.2.1

Group the first two terms and the last two terms.

$(3 x^{2} - 2 x) - 12 x + 8 = 0$

Step 4.4.2.2

Factor out the greatest common factor (GCF) from each group.

$x (3 x - 2) - 4 (3 x - 2) = 0$

Step 4.4.3

Factor the polynomial by factoring out the greatest common factor,

$3 x - 2$ .

$(3 x - 2) (x - 4) = 0$

Step 4.5

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$3 x - 2 = 0$

$x - 4 = 0$

Step 4.6

Set

$3 x - 2$ equal to

$0$ and solve for

$x$ .

Tap for more steps...

Step 4.6.1

Set

$3 x - 2$ equal to

$0$ .

$3 x - 2 = 0$

Step 4.6.2

Solve

$3 x - 2 = 0$ for

$x$ .

Tap for more steps...

Step 4.6.2.1

Add

$2$ to both sides of the equation.

$3 x = 2$

Step 4.6.2.2

Divide each term in

$3 x = 2$ by

$3$ and simplify.

Tap for more steps...

Step 4.6.2.2.1

Divide each term in

$3 x = 2$ by

$3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Step 4.6.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.6.2.2.2.1

Cancel the common factor of

$3$ .

Tap for more steps...

Step 4.6.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Step 4.6.2.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{2}{3}$

Step 4.7

Set

$x - 4$ equal to

$0$ and solve for

$x$ .

Tap for more steps...

Step 4.7.1

Set

$x - 4$ equal to

$0$ .

$x - 4 = 0$

Step 4.7.2

Add

$4$ to both sides of the equation.

$x = 4$

Step 4.8

The final solution is all the values that make

$(3 x - 2) (x - 4) = 0$ true.

$x = \frac{2}{3}, 4$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{3}, 4$

Decimal Form:

$x = 0.\overline{6}, 4$