Algebra Examples

$f (x) = - \frac{2}{2 x - 4} + 1$

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 $x$ .

$0 = - \frac{2}{2 x - 4} + 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- \frac{2}{2 x - 4} + 1 = 0$ .

$- \frac{2}{2 x - 4} + 1 = 0$

Step 1.2.2

Subtract $1$ from both sides of the equation.

$- \frac{2}{2 x - 4} = - 1$

Step 1.2.3

Factor each term.

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Step 1.2.3.1

Factor $2$ out of $2 x - 4$ .

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Step 1.2.3.1.1

Factor $2$ out of $2 x$ .

$- \frac{2}{2 (x) - 4} = - 1$

Step 1.2.3.1.2

Factor $2$ out of $- 4$ .

$- \frac{2}{2 x + 2 \cdot - 2} = - 1$

Step 1.2.3.1.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

$- \frac{2}{2 (x - 2)} = - 1$

Step 1.2.3.2

Reduce the expression $\frac{2}{2 (x - 2)}$ by cancelling the common factors.

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Step 1.2.3.2.1

Cancel the common factor.

$- \frac{2}{2 (x - 2)} = - 1$

Step 1.2.3.2.2

Rewrite the expression.

$- \frac{1}{x - 2} = - 1$

Step 1.2.4

Find the LCD of the terms in the equation.

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Step 1.2.4.1

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

$x - 2, 1$

Step 1.2.4.2

Remove parentheses.

$x - 2, 1$

Step 1.2.4.3

The LCM of one and any expression is the expression.

$x - 2$

Step 1.2.5

Multiply each term in $- \frac{1}{x - 2} = - 1$ by $x - 2$ to eliminate the fractions.

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Step 1.2.5.1

Multiply each term in $- \frac{1}{x - 2} = - 1$ by $x - 2$ .

$- \frac{1}{x - 2} (x - 2) = - (x - 2)$

Step 1.2.5.2

Simplify the left side.

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Step 1.2.5.2.1

Cancel the common factor of $x - 2$ .

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Step 1.2.5.2.1.1

Move the leading negative in $- \frac{1}{x - 2}$ into the numerator.

$\frac{- 1}{x - 2} (x - 2) = - (x - 2)$

Step 1.2.5.2.1.2

Cancel the common factor.

$\frac{- 1}{x - 2} (x - 2) = - (x - 2)$

Step 1.2.5.2.1.3

Rewrite the expression.

$- 1 = - (x - 2)$

Step 1.2.5.3

Simplify the right side.

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Step 1.2.5.3.1

Apply the distributive property.

$- 1 = - x - - 2$

Step 1.2.5.3.2

Multiply $- 1$ by $- 2$ .

$- 1 = - x + 2$

Step 1.2.6

Solve the equation.

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Step 1.2.6.1

Rewrite the equation as $- x + 2 = - 1$ .

$- x + 2 = - 1$

Step 1.2.6.2

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

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Step 1.2.6.2.1

Subtract $2$ from both sides of the equation.

$- x = - 1 - 2$

Step 1.2.6.2.2

Subtract $2$ from $- 1$ .

$- x = - 3$

Step 1.2.6.3

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 1.2.6.3.1

Divide each term in $- x = - 3$ by $- 1$ .

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

Step 1.2.6.3.2

Simplify the left side.

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Step 1.2.6.3.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.6.3.2.2

Divide $x$ by $1$ .

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

Step 1.2.6.3.3

Simplify the right side.

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Step 1.2.6.3.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(3, 0)$

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 $y$ .

$y = - \frac{2}{2 (0) - 4} + 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - \frac{2}{2 (0) - 4} + 1$

Step 2.2.2

Simplify $- \frac{2}{2 (0) - 4} + 1$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Cancel the common factor of $2$ and $2 (0) - 4$ .

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Step 2.2.2.1.1.1

Factor $2$ out of $2$ .

$y = - \frac{2 (1)}{2 (0) - 4} + 1$

Step 2.2.2.1.1.2

Cancel the common factors.

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Step 2.2.2.1.1.2.1

Factor $2$ out of $- 4$ .

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

Step 2.2.2.1.1.2.2

Factor $2$ out of $2 (0) + 2 (- 2)$ .

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

Step 2.2.2.1.1.2.3

Cancel the common factor.

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

Step 2.2.2.1.1.2.4

Rewrite the expression.

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

Step 2.2.2.1.2

Subtract $2$ from $0$ .

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

Step 2.2.2.1.3

Move the negative in front of the fraction.

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

Step 2.2.2.1.4

Multiply $- - \frac{1}{2}$ .

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Step 2.2.2.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.2.1.4.2

Multiply $\frac{1}{2}$ by $1$ .

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

Step 2.2.2.2

Simplify the expression.

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

Combine the numerators over the common denominator.

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

Step 2.2.2.2.3

Add $1$ and $2$ .

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

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{3}{2})$

Step 3

List the intersections.

x-intercept(s): $(3, 0)$

y-intercept(s): $(0, \frac{3}{2})$

Step 4