Calculus Examples

$\frac{3 x}{- 4 x + 4} \geq 8$

Step 1

Multiply both sides by $- 4 x + 4$ .

$\frac{3 x}{- 4 x + 4} (- 4 x + 4) = 8 (- 4 x + 4)$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

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

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Step 2.1.1.1

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

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Step 2.1.1.1.1

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

$\frac{3 x}{4 (- x) + 4} (- 4 x + 4) = 8 (- 4 x + 4)$

Step 2.1.1.1.2

Factor $4$ out of $4$ .

$\frac{3 x}{4 (- x) + 4 (1)} (- 4 x + 4) = 8 (- 4 x + 4)$

Step 2.1.1.1.3

Factor $4$ out of $4 (- x) + 4 (1)$ .

$\frac{3 x}{4 (- x + 1)} (- 4 x + 4) = 8 (- 4 x + 4)$

Step 2.1.1.2

Multiply $\frac{3 x}{4 (- x + 1)}$ by $- 4 x + 4$ .

$\frac{3 x (- 4 x + 4)}{4 (- x + 1)} = 8 (- 4 x + 4)$

Step 2.1.1.3

Cancel the common factor of $- 4 x + 4$ and $4$ .

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Step 2.1.1.3.1

Factor $4$ out of $3 x (- 4 x + 4)$ .

$\frac{4 (3 x (- x + 1))}{4 (- x + 1)} = 8 (- 4 x + 4)$

Step 2.1.1.3.2

Cancel the common factors.

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Step 2.1.1.3.2.1

Cancel the common factor.

$\frac{4 (3 x (- x + 1))}{4 (- x + 1)} = 8 (- 4 x + 4)$

Step 2.1.1.3.2.2

Rewrite the expression.

$\frac{3 x (- x + 1)}{- x + 1} = 8 (- 4 x + 4)$

Step 2.1.1.4

Cancel the common factor of $- x + 1$ .

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Step 2.1.1.4.1

Cancel the common factor.

$\frac{3 x (- x + 1)}{- x + 1} = 8 (- 4 x + 4)$

Step 2.1.1.4.2

Divide $3 x$ by $1$ .

$3 x = 8 (- 4 x + 4)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $8 (- 4 x + 4)$ .

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Step 2.2.1.1

Apply the distributive property.

$3 x = 8 (- 4 x) + 8 \cdot 4$

Step 2.2.1.2

Multiply.

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Step 2.2.1.2.1

Multiply $- 4$ by $8$ .

$3 x = - 32 x + 8 \cdot 4$

Step 2.2.1.2.2

Multiply $8$ by $4$ .

$3 x = - 32 x + 32$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Add $32 x$ to both sides of the equation.

$3 x + 32 x = 32$

Step 3.1.2

Add $3 x$ and $32 x$ .

$35 x = 32$

Step 3.2

Divide each term in $35 x = 32$ by $35$ and simplify.

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Step 3.2.1

Divide each term in $35 x = 32$ by $35$ .

$\frac{35 x}{35} = \frac{32}{35}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $35$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{35 x}{35} = \frac{32}{35}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{32}{35}$

Step 4

Find the domain of $\frac{3 x}{4 (- x + 1)} - 8$ .

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Step 4.1

Set the denominator in $\frac{3 x}{4 (- x + 1)}$ equal to $0$ to find where the expression is undefined.

$4 (- x + 1) = 0$

Step 4.2

Solve for $x$ .

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Step 4.2.1

Divide each term in $4 (- x + 1) = 0$ by $4$ and simplify.

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Step 4.2.1.1

Divide each term in $4 (- x + 1) = 0$ by $4$ .

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

Step 4.2.1.2

Simplify the left side.

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Step 4.2.1.2.1

Cancel the common factor of $4$ .

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Step 4.2.1.2.1.1

Cancel the common factor.

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

Step 4.2.1.2.1.2

Divide $- x + 1$ by $1$ .

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

Step 4.2.1.3

Simplify the right side.

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Step 4.2.1.3.1

Divide $0$ by $4$ .

$- x + 1 = 0$

Step 4.2.2

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 4.2.3

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

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Step 4.2.3.1

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

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

Step 4.2.3.2

Simplify the left side.

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Step 4.2.3.2.1

Dividing two negative values results in a positive value.

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

Step 4.2.3.2.2

Divide $x$ by $1$ .

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

Step 4.2.3.3

Simplify the right side.

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Step 4.2.3.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 4.3

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

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

Step 5

Use each root to create test intervals.

$x < \frac{32}{35}$

$\frac{32}{35} < x < 1$

$x > 1$

Step 6

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

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Step 6.1

Test a value on the interval $x < \frac{32}{35}$ to see if it makes the inequality true.

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Step 6.1.1

Choose a value on the interval $x < \frac{32}{35}$ and see if this value makes the original inequality true.

$x = 0$

Step 6.1.2

Replace $x$ with $0$ in the original inequality.

$\frac{3 (0)}{- 4 \cdot 0 + 4} \geq 8$

Step 6.1.3

The left side $0$ is less than the right side $8$ , which means that the given statement is false.

False

Step 6.2

Test a value on the interval $\frac{32}{35} < x < 1$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $\frac{32}{35} < x < 1$ and see if this value makes the original inequality true.

$x = 0.96$

Step 6.2.2

Replace $x$ with $0.96$ in the original inequality.

$\frac{3 (0.96)}{- 4 \cdot 0.96 + 4} \geq 8$

Step 6.2.3

The left side $18$ is greater than the right side $8$ , which means that the given statement is always true.

True

Step 6.3

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

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Step 6.3.1

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

$x = 4$

Step 6.3.2

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

$\frac{3 (4)}{- 4 \cdot 4 + 4} \geq 8$

Step 6.3.3

The left side $- 1$ is less than the right side $8$ , which means that the given statement is false.

False

Step 6.4

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

$x < \frac{32}{35}$ False

$\frac{32}{35} < x < 1$ True

$x > 1$ False

$x < \frac{32}{35}$ False

$\frac{32}{35} < x < 1$ True

$x > 1$ False

Step 7

The solution consists of all of the true intervals.

$\frac{32}{35} \leq x < 1$

Step 8

Convert the inequality to interval notation.

$[\frac{32}{35}, 1)$

Step 9