Algebra Examples

$\frac{1}{x - 3} + 4$

Step 1

Write $\frac{1}{x - 3} + 4$ as an equation.

$y = \frac{1}{x - 3} + 4$

Step 2

Find the x-intercepts.

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Step 2.1

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

$0 = \frac{1}{x - 3} + 4$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $\frac{1}{x - 3} + 4 = 0$ .

$\frac{1}{x - 3} + 4 = 0$

Step 2.2.2

Subtract $4$ from both sides of the equation.

$\frac{1}{x - 3} = - 4$

Step 2.2.3

Find the LCD of the terms in the equation.

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Step 2.2.3.1

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

$x - 3, 1$

Step 2.2.3.2

Remove parentheses.

$x - 3, 1$

Step 2.2.3.3

The LCM of one and any expression is the expression.

$x - 3$

Step 2.2.4

Multiply each term in $\frac{1}{x - 3} = - 4$ by $x - 3$ to eliminate the fractions.

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Step 2.2.4.1

Multiply each term in $\frac{1}{x - 3} = - 4$ by $x - 3$ .

$\frac{1}{x - 3} (x - 3) = - 4 (x - 3)$

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

Cancel the common factor of $x - 3$ .

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Step 2.2.4.2.1.1

Cancel the common factor.

$\frac{1}{x - 3} (x - 3) = - 4 (x - 3)$

Step 2.2.4.2.1.2

Rewrite the expression.

$1 = - 4 (x - 3)$

Step 2.2.4.3

Simplify the right side.

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Step 2.2.4.3.1

Apply the distributive property.

$1 = - 4 x - 4 \cdot - 3$

Step 2.2.4.3.2

Multiply $- 4$ by $- 3$ .

$1 = - 4 x + 12$

Step 2.2.5

Solve the equation.

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Step 2.2.5.1

Rewrite the equation as $- 4 x + 12 = 1$ .

$- 4 x + 12 = 1$

Step 2.2.5.2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.2.5.2.1

Subtract $12$ from both sides of the equation.

$- 4 x = 1 - 12$

Step 2.2.5.2.2

Subtract $12$ from $1$ .

$- 4 x = - 11$

Step 2.2.5.3

Divide each term in $- 4 x = - 11$ by $- 4$ and simplify.

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Step 2.2.5.3.1

Divide each term in $- 4 x = - 11$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 11}{- 4}$

Step 2.2.5.3.2

Simplify the left side.

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Step 2.2.5.3.2.1

Cancel the common factor of $- 4$ .

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Step 2.2.5.3.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 11}{- 4}$

Step 2.2.5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 11}{- 4}$

Step 2.2.5.3.3

Simplify the right side.

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Step 2.2.5.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{11}{4}$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{11}{4}, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = \frac{1}{(0) - 3} + 4$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = \frac{1}{0 - 3} + 4$

Step 3.2.2

Remove parentheses.

$y = \frac{1}{(0) - 3} + 4$

Step 3.2.3

Simplify $\frac{1}{(0) - 3} + 4$ .

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Subtract $3$ from $0$ .

$y = \frac{1}{- 3} + 4$

Step 3.2.3.1.2

Move the negative in front of the fraction.

$y = - \frac{1}{3} + 4$

Step 3.2.3.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{1}{3} + 4 \cdot \frac{3}{3}$

Step 3.2.3.3

Combine $4$ and $\frac{3}{3}$ .

$y = - \frac{1}{3} + \frac{4 \cdot 3}{3}$

Step 3.2.3.4

Combine the numerators over the common denominator.

$y = \frac{- 1 + 4 \cdot 3}{3}$

Step 3.2.3.5

Simplify the numerator.

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Step 3.2.3.5.1

Multiply $4$ by $3$ .

$y = \frac{- 1 + 12}{3}$

Step 3.2.3.5.2

Add $- 1$ and $12$ .

$y = \frac{11}{3}$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{11}{3})$

Step 4

List the intersections.

x-intercept(s): $(\frac{11}{4}, 0)$

y-intercept(s): $(0, \frac{11}{3})$

Step 5