Precalculus Examples

$\frac{6}{x - 1} - \frac{6}{x} \geq 1$

Step 1

Simplify $\frac{6}{x - 1} - \frac{6}{x}$ .

Tap for more steps...

Step 1.1

To write $\frac{6}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{6}{x - 1} \cdot \frac{x}{x} - \frac{6}{x} \geq 1$

Step 1.2

To write $- \frac{6}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{6}{x - 1} \cdot \frac{x}{x} - \frac{6}{x} \cdot \frac{x - 1}{x - 1} \geq 1$

Step 1.3

Write each expression with a common denominator of $(x - 1) x$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.3.1

Multiply $\frac{6}{x - 1}$ by $\frac{x}{x}$ .

$\frac{6 x}{(x - 1) x} - \frac{6}{x} \cdot \frac{x - 1}{x - 1} \geq 1$

Step 1.3.2

Multiply $\frac{6}{x}$ by $\frac{x - 1}{x - 1}$ .

$\frac{6 x}{(x - 1) x} - \frac{6 (x - 1)}{x (x - 1)} \geq 1$

Step 1.3.3

Reorder the factors of $(x - 1) x$ .

$\frac{6 x}{x (x - 1)} - \frac{6 (x - 1)}{x (x - 1)} \geq 1$

Step 1.4

Combine the numerators over the common denominator.

$\frac{6 x - 6 (x - 1)}{x (x - 1)} \geq 1$

Step 1.5

Simplify the numerator.

Tap for more steps...

Step 1.5.1

Factor $6$ out of $6 x - 6 (x - 1)$ .

Tap for more steps...

Step 1.5.1.1

Factor $6$ out of $6 x$ .

$\frac{6 (x) - 6 (x - 1)}{x (x - 1)} \geq 1$

Step 1.5.1.2

Factor $6$ out of $- 6 (x - 1)$ .

$\frac{6 (x) + 6 (- (x - 1))}{x (x - 1)} \geq 1$

Step 1.5.1.3

Factor $6$ out of $6 (x) + 6 (- (x - 1))$ .

$\frac{6 (x - (x - 1))}{x (x - 1)} \geq 1$

Step 1.5.2

Apply the distributive property.

$\frac{6 (x - x - - 1)}{x (x - 1)} \geq 1$

Step 1.5.3

Multiply $- 1$ by $- 1$ .

$\frac{6 (x - x + 1)}{x (x - 1)} \geq 1$

Step 1.5.4

Subtract $x$ from $x$ .

$\frac{6 (0 + 1)}{x (x - 1)} \geq 1$

Step 1.5.5

Add $0$ and $1$ .

$\frac{6 \cdot 1}{x (x - 1)} \geq 1$

Step 1.6

Multiply $6$ by $1$ .

$\frac{6}{x (x - 1)} \geq 1$

Step 2

Multiply both sides by $x (x - 1)$ .

$\frac{6}{x (x - 1)} (x (x - 1)) = 1 (x (x - 1))$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Simplify the left side.

Tap for more steps...

Step 3.1.1

Cancel the common factor of $x (x - 1)$ .

Tap for more steps...

Step 3.1.1.1

Cancel the common factor.

$\frac{6}{x (x - 1)} (x (x - 1)) = 1 (x (x - 1))$

Step 3.1.1.2

Rewrite the expression.

$6 = 1 (x (x - 1))$

Step 3.2

Simplify the right side.

Tap for more steps...

Step 3.2.1

Simplify $1 (x (x - 1))$ .

Tap for more steps...

Step 3.2.1.1

Simplify by multiplying through.

Tap for more steps...

Step 3.2.1.1.1

Multiply $x (x - 1)$ by $1$ .

$6 = x (x - 1)$

Step 3.2.1.1.2

Apply the distributive property.

$6 = x \cdot x + x \cdot - 1$

Step 3.2.1.1.3

Simplify the expression.

Tap for more steps...

Step 3.2.1.1.3.1

Multiply $x$ by $x$ .

$6 = x^{2} + x \cdot - 1$

Step 3.2.1.1.3.2

Move $- 1$ to the left of $x$ .

$6 = x^{2} - 1 \cdot x$

Step 3.2.1.2

Rewrite $- 1 x$ as $- x$ .

$6 = x^{2} - x$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $x^{2} - x = 6$ .

$x^{2} - x = 6$

Step 4.2

Subtract $6$ from both sides of the equation.

$x^{2} - x - 6 = 0$

Step 4.3

Factor $x^{2} - x - 6$ using the AC method.

Tap for more steps...

Step 4.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 4.3.2

Write the factored form using these integers.

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

Step 4.4

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

$x - 3 = 0$

$x + 2 = 0$

Step 4.5

Set $x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4.6

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

Tap for more steps...

Step 4.6.1

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

$x + 2 = 0$

Step 4.6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4.7

The final solution is all the values that make $(x - 3) (x + 2) = 0$ true.

$x = 3, - 2$

Step 5

Find the domain of $\frac{6}{x (x - 1)} - 1$ .

Tap for more steps...

Step 5.1

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

$x (x - 1) = 0$

Step 5.2

Solve for $x$ .

Tap for more steps...

Step 5.2.1

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

$x = 0$

$x - 1 = 0$

Step 5.2.2

Set $x$ equal to $0$ .

$x = 0$

Step 5.2.3

Set $x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.2.3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 5.2.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 5.2.4

The final solution is all the values that make $x (x - 1) = 0$ true.

$x = 0, 1$

Step 5.3

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

$(- \infty, 0) \cup (0, 1) \cup (1, \infty)$

Step 6

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 0$

$0 < x < 1$

$1 < x < 3$

$x > 3$

Step 7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 7.1

Test a value on the interval $x < - 2$ to see if it makes the inequality true.

Tap for more steps...

Step 7.1.1

Choose a value on the interval $x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

Step 7.1.2

Replace $x$ with $- 4$ in the original inequality.

$\frac{6}{(- 4) - 1} - \frac{6}{- 4} \geq 1$

Step 7.1.3

The left side $0.3$ is less than the right side $1$ , which means that the given statement is false.

False

Step 7.2

Test a value on the interval $- 2 < x < 0$ to see if it makes the inequality true.

Tap for more steps...

Step 7.2.1

Choose a value on the interval $- 2 < x < 0$ and see if this value makes the original inequality true.

$x = - 1$

Step 7.2.2

Replace $x$ with $- 1$ in the original inequality.

$\frac{6}{(- 1) - 1} - \frac{6}{- 1} \geq 1$

Step 7.2.3

The left side $3$ is greater than the right side $1$ , which means that the given statement is always true.

True

Step 7.3

Test a value on the interval $0 < x < 1$ to see if it makes the inequality true.

Tap for more steps...

Step 7.3.1

Choose a value on the interval $0 < x < 1$ and see if this value makes the original inequality true.

$x = 0.5$

Step 7.3.2

Replace $x$ with $0.5$ in the original inequality.

$\frac{6}{(0.5) - 1} - \frac{6}{0.5} \geq 1$

Step 7.3.3

The left side $- 24$ is less than the right side $1$ , which means that the given statement is false.

False

Step 7.4

Test a value on the interval $1 < x < 3$ to see if it makes the inequality true.

Tap for more steps...

Step 7.4.1

Choose a value on the interval $1 < x < 3$ and see if this value makes the original inequality true.

$x = 2$

Step 7.4.2

Replace $x$ with $2$ in the original inequality.

$\frac{6}{(2) - 1} - \frac{6}{2} \geq 1$

Step 7.4.3

The left side $3$ is greater than the right side $1$ , which means that the given statement is always true.

True

Step 7.5

Test a value on the interval $x > 3$ to see if it makes the inequality true.

Tap for more steps...

Step 7.5.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 7.5.2

Replace $x$ with $6$ in the original inequality.

$\frac{6}{(6) - 1} - \frac{6}{6} \geq 1$

Step 7.5.3

The left side $0.2$ is less than the right side $1$ , which means that the given statement is false.

False

Step 7.6

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 2$ False

$- 2 < x < 0$ True

$0 < x < 1$ False

$1 < x < 3$ True

$x > 3$ False

$x < - 2$ False

$- 2 < x < 0$ True

$0 < x < 1$ False

$1 < x < 3$ True

$x > 3$ False

Step 8

The solution consists of all of the true intervals.

$- 2 \leq x < 0$ or $1 < x \leq 3$

Step 9

Convert the inequality to interval notation.

$[- 2, 0) \cup (1, 3]$

Step 10