Precalculus Examples

$y = \frac{2}{x + 7} + \frac{3}{3 - x}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{2}{x + 7} + \frac{3}{3 - x}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{2}{x + 7} + \frac{3}{3 - x} = 0$ .

$\frac{2}{x + 7} + \frac{3}{3 - x} = 0$

Step 1.2.2

Find the LCD of the terms in the equation.

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Step 1.2.2.1

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

$x + 7, 3 - x, 1$

Step 1.2.2.2

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 1.2.2.3

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

Not prime

Step 1.2.2.4

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 1.2.2.5

The factor for $x + 7$ is $x + 7$ itself.

$(x + 7) = x + 7$

$(x + 7)$ occurs $1$ time.

Step 1.2.2.6

The factor for $3 - x$ is $3 - x$ itself.

$(3 - x) = 3 - x$

$(3 - x)$ occurs $1$ time.

Step 1.2.2.7

The LCM of $x + 7, 3 - x$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(x + 7) (3 - x)$

Step 1.2.3

Multiply each term in $\frac{2}{x + 7} + \frac{3}{3 - x} = 0$ by $(x + 7) (3 - x)$ to eliminate the fractions.

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Step 1.2.3.1

Multiply each term in $\frac{2}{x + 7} + \frac{3}{3 - x} = 0$ by $(x + 7) (3 - x)$ .

$\frac{2}{x + 7} ((x + 7) (3 - x)) + \frac{3}{3 - x} ((x + 7) (3 - x)) = 0 ((x + 7) (3 - x))$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Simplify each term.

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Step 1.2.3.2.1.1

Cancel the common factor of $x + 7$ .

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Step 1.2.3.2.1.1.1

Cancel the common factor.

$\frac{2}{x + 7} ((x + 7) (3 - x)) + \frac{3}{3 - x} ((x + 7) (3 - x)) = 0 ((x + 7) (3 - x))$

Step 1.2.3.2.1.1.2

Rewrite the expression.

$2 (3 - x) + \frac{3}{3 - x} ((x + 7) (3 - x)) = 0 ((x + 7) (3 - x))$

Step 1.2.3.2.1.2

Apply the distributive property.

$2 \cdot 3 + 2 (- x) + \frac{3}{3 - x} ((x + 7) (3 - x)) = 0 ((x + 7) (3 - x))$

Step 1.2.3.2.1.3

Multiply $2$ by $3$ .

$6 + 2 (- x) + \frac{3}{3 - x} ((x + 7) (3 - x)) = 0 ((x + 7) (3 - x))$

Step 1.2.3.2.1.4

Multiply $- 1$ by $2$ .

$6 - 2 x + \frac{3}{3 - x} ((x + 7) (3 - x)) = 0 ((x + 7) (3 - x))$

Step 1.2.3.2.1.5

Cancel the common factor of $3 - x$ .

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Step 1.2.3.2.1.5.1

Factor $3 - x$ out of $(x + 7) (3 - x)$ .

$6 - 2 x + \frac{3}{3 - x} ((3 - x) (x + 7)) = 0 ((x + 7) (3 - x))$

Step 1.2.3.2.1.5.2

Cancel the common factor.

$6 - 2 x + \frac{3}{3 - x} ((3 - x) (x + 7)) = 0 ((x + 7) (3 - x))$

Step 1.2.3.2.1.5.3

Rewrite the expression.

$6 - 2 x + 3 (x + 7) = 0 ((x + 7) (3 - x))$

Step 1.2.3.2.1.6

Apply the distributive property.

$6 - 2 x + 3 x + 3 \cdot 7 = 0 ((x + 7) (3 - x))$

Step 1.2.3.2.1.7

Multiply $3$ by $7$ .

$6 - 2 x + 3 x + 21 = 0 ((x + 7) (3 - x))$

Step 1.2.3.2.2

Simplify by adding terms.

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Step 1.2.3.2.2.1

Add $6$ and $21$ .

$- 2 x + 3 x + 27 = 0 ((x + 7) (3 - x))$

Step 1.2.3.2.2.2

Add $- 2 x$ and $3 x$ .

$x + 27 = 0 ((x + 7) (3 - x))$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

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

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Step 1.2.3.3.1.1

Apply the distributive property.

$x + 27 = 0 (x (3 - x) + 7 (3 - x))$

Step 1.2.3.3.1.2

Apply the distributive property.

$x + 27 = 0 (x \cdot 3 + x (- x) + 7 (3 - x))$

Step 1.2.3.3.1.3

Apply the distributive property.

$x + 27 = 0 (x \cdot 3 + x (- x) + 7 \cdot 3 + 7 (- x))$

Step 1.2.3.3.2

Simplify and combine like terms.

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Step 1.2.3.3.2.1

Simplify each term.

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Step 1.2.3.3.2.1.1

Move $3$ to the left of $x$ .

$x + 27 = 0 (3 \cdot x + x (- x) + 7 \cdot 3 + 7 (- x))$

Step 1.2.3.3.2.1.2

Rewrite using the commutative property of multiplication.

$x + 27 = 0 (3 x - x \cdot x + 7 \cdot 3 + 7 (- x))$

Step 1.2.3.3.2.1.3

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

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Step 1.2.3.3.2.1.3.1

Move $x$ .

$x + 27 = 0 (3 x - (x \cdot x) + 7 \cdot 3 + 7 (- x))$

Step 1.2.3.3.2.1.3.2

Multiply $x$ by $x$ .

$x + 27 = 0 (3 x - x^{2} + 7 \cdot 3 + 7 (- x))$

Step 1.2.3.3.2.1.4

Multiply $7$ by $3$ .

$x + 27 = 0 (3 x - x^{2} + 21 + 7 (- x))$

Step 1.2.3.3.2.1.5

Multiply $- 1$ by $7$ .

$x + 27 = 0 (3 x - x^{2} + 21 - 7 x)$

Step 1.2.3.3.2.2

Subtract $7 x$ from $3 x$ .

$x + 27 = 0 (- 4 x - x^{2} + 21)$

Step 1.2.3.3.3

Multiply $0$ by $- 4 x - x^{2} + 21$ .

$x + 27 = 0$

Step 1.2.4

Subtract $27$ from both sides of the equation.

$x = - 27$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 27, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{2}{(0) + 7} + \frac{3}{3 - (0)}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{2}{0 + 7} + \frac{3}{3 - (0)}$

Step 2.2.2

Remove parentheses.

$y = \frac{2}{(0) + 7} + \frac{3}{3 - (0)}$

Step 2.2.3

Simplify $\frac{2}{(0) + 7} + \frac{3}{3 - (0)}$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Add $0$ and $7$ .

$y = \frac{2}{7} + \frac{3}{3 - (0)}$

Step 2.2.3.1.2

Cancel the common factor of $3$ and $3 - (0)$ .

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Step 2.2.3.1.2.1

Rewrite $3$ as $- 1 (- 3)$ .

$y = \frac{2}{7} + \frac{3}{- 1 (- 3) - (0)}$

Step 2.2.3.1.2.2

Factor $- 1$ out of $- 1 (- 3) - (0)$ .

$y = \frac{2}{7} + \frac{3}{- 1 (- 3 + 0)}$

Step 2.2.3.1.2.3

Factor $3$ out of $3$ .

$y = \frac{2}{7} + \frac{3 (1)}{- 1 (- 3 + 0)}$

Step 2.2.3.1.2.4

Cancel the common factors.

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Step 2.2.3.1.2.4.1

Factor $3$ out of $- 1 (- 3 + 0)$ .

$y = \frac{2}{7} + \frac{3 (1)}{3 (- 1 (- 1 + 0))}$

Step 2.2.3.1.2.4.2

Cancel the common factor.

$y = \frac{2}{7} + \frac{3 \cdot 1}{3 (- 1 (- 1 + 0))}$

Step 2.2.3.1.2.4.3

Rewrite the expression.

$y = \frac{2}{7} + \frac{1}{- 1 (- 1 + 0)}$

Step 2.2.3.1.3

Cancel the common factor of $1$ and $- 1$ .

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Step 2.2.3.1.3.1

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

$y = \frac{2}{7} + \frac{- 1 (- 1)}{- 1 (- 1 + 0)}$

Step 2.2.3.1.3.2

Move the negative in front of the fraction.

$y = \frac{2}{7} - \frac{1}{- 1 + 0}$

Step 2.2.3.1.4

Add $- 1$ and $0$ .

$y = \frac{2}{7} - \frac{1}{- 1}$

Step 2.2.3.1.5

Divide $1$ by $- 1$ .

$y = \frac{2}{7} - - 1$

Step 2.2.3.1.6

Multiply $- 1$ by $- 1$ .

$y = \frac{2}{7} + 1$

Step 2.2.3.2

Simplify the expression.

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Step 2.2.3.2.1

Write $1$ as a fraction with a common denominator.

$y = \frac{2}{7} + \frac{7}{7}$

Step 2.2.3.2.2

Combine the numerators over the common denominator.

$y = \frac{2 + 7}{7}$

Step 2.2.3.2.3

Add $2$ and $7$ .

$y = \frac{9}{7}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{9}{7})$

Step 3

List the intersections.

x-intercept(s): $(- 27, 0)$

y-intercept(s): $(0, \frac{9}{7})$

Step 4