Linear Algebra Examples

$\frac{3 r - 9}{4 r^{2} + 8 r} \cdot \frac{2 r + 4}{18 - 6 r}$

Step 1

Set the denominator in $\frac{3 r - 9}{4 r^{2} + 8 r}$ equal to $0$ to find where the expression is undefined.

$4 r^{2} + 8 r = 0$

Step 2

Solve for $r$ .

Tap for more steps...

Step 2.1

Factor $4 r$ out of $4 r^{2} + 8 r$ .

Tap for more steps...

Step 2.1.1

Factor $4 r$ out of $4 r^{2}$ .

$4 r (r) + 8 r = 0$

Step 2.1.2

Factor $4 r$ out of $8 r$ .

$4 r (r) + 4 r (2) = 0$

Step 2.1.3

Factor $4 r$ out of $4 r (r) + 4 r (2)$ .

$4 r (r + 2) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$r = 0$

$r + 2 = 0$

Step 2.3

Set $r$ equal to $0$ .

$r = 0$

Step 2.4

Set $r + 2$ equal to $0$ and solve for $r$ .

Tap for more steps...

Step 2.4.1

Set $r + 2$ equal to $0$ .

$r + 2 = 0$

Step 2.4.2

Subtract $2$ from both sides of the equation.

$r = - 2$

Step 2.5

The final solution is all the values that make $4 r (r + 2) = 0$ true.

$r = 0, - 2$

Step 3

Set the denominator in $\frac{2 r + 4}{18 - 6 r}$ equal to $0$ to find where the expression is undefined.

$18 - 6 r = 0$

Step 4

Solve for $r$ .

Tap for more steps...

Step 4.1

Subtract $18$ from both sides of the equation.

$- 6 r = - 18$

Step 4.2

Divide each term in $- 6 r = - 18$ by $- 6$ and simplify.

Tap for more steps...

Step 4.2.1

Divide each term in $- 6 r = - 18$ by $- 6$ .

$\frac{- 6 r}{- 6} = \frac{- 18}{- 6}$

Step 4.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.1

Cancel the common factor of $- 6$ .

Tap for more steps...

Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 6 r}{- 6} = \frac{- 18}{- 6}$

Step 4.2.2.1.2

Divide $r$ by $1$ .

$r = \frac{- 18}{- 6}$

Step 4.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.3.1

Divide $- 18$ by $- 6$ .

$r = 3$

Step 5

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

Interval Notation:

$(- \infty, - 2) \cup (- 2, 0) \cup (0, 3) \cup (3, \infty)$

Set-Builder Notation:

${r | r \neq 0, - 2, 3}$

Step 6