Precalculus Examples

$y = \frac{6 + 9 x}{6 - | x - 1 |}$

Step 1

Set the denominator in $\frac{6 + 9 x}{6 - | x - 1 |}$ equal to $0$ to find where the expression is undefined.

$6 - | x - 1 | = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $6$ from both sides of the equation.

$- | x - 1 | = - 6$

Step 2.2

Divide each term in $- | x - 1 | = - 6$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- | x - 1 | = - 6$ by $- 1$ .

$\frac{- | x - 1 |}{- 1} = \frac{- 6}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{| x - 1 |}{1} = \frac{- 6}{- 1}$

Step 2.2.2.2

Divide $| x - 1 |$ by $1$ .

$| x - 1 | = \frac{- 6}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 6$ by $- 1$ .

$| x - 1 | = 6$

Step 2.3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x - 1 = \pm 6$

Step 2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x - 1 = 6$

Step 2.4.2

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

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Step 2.4.2.1

Add $1$ to both sides of the equation.

$x = 6 + 1$

Step 2.4.2.2

Add $6$ and $1$ .

$x = 7$

Step 2.4.3

Next, use the negative value of the $\pm$ to find the second solution.

$x - 1 = - 6$

Step 2.4.4

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

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Step 2.4.4.1

Add $1$ to both sides of the equation.

$x = - 6 + 1$

Step 2.4.4.2

Add $- 6$ and $1$ .

$x = - 5$

Step 2.4.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = 7, - 5$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 5) \cup (- 5, 7) \cup (7, \infty)$

Set-Builder Notation:

${x | x \neq - 5, 7}$

Step 4