Calculus Examples

${(x - 2)}^{2} (24 - 4 x) > 0$

Step 1

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

${(x - 2)}^{2} = 0$

$24 - 4 x = 0$

Step 2

Set ${(x - 2)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.1

Set ${(x - 2)}^{2}$ equal to $0$ .

${(x - 2)}^{2} = 0$

Step 2.2

Solve ${(x - 2)}^{2} = 0$ for $x$ .

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Step 2.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3

Set $24 - 4 x$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $24 - 4 x$ equal to $0$ .

$24 - 4 x = 0$

Step 3.2

Solve $24 - 4 x = 0$ for $x$ .

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Step 3.2.1

Subtract $24$ from both sides of the equation.

$- 4 x = - 24$

Step 3.2.2

Divide each term in $- 4 x = - 24$ by $- 4$ and simplify.

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Step 3.2.2.1

Divide each term in $- 4 x = - 24$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 24}{- 4}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $- 4$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 24}{- 4}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 24}{- 4}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $- 24$ by $- 4$ .

$x = 6$

Step 4

The final solution is all the values that make ${(x - 2)}^{2} (24 - 4 x) > 0$ true.

$x = 2, 6$

Step 5

Use each root to create test intervals.

$x < 2$

$2 < x < 6$

$x > 6$

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 $x < 2$ to see if it makes the inequality true.

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Step 6.1.1

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

$x = 0$

Step 6.1.2

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

${((0) - 2)}^{2} (24 - 4 \cdot 0) > 0$

Step 6.1.3

The left side $96$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.2

Test a value on the interval $2 < x < 6$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $2 < x < 6$ and see if this value makes the original inequality true.

$x = 4$

Step 6.2.2

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

${((4) - 2)}^{2} (24 - 4 \cdot 4) > 0$

Step 6.2.3

The left side $32$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.3

Test a value on the interval $x > 6$ to see if it makes the inequality true.

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Step 6.3.1

Choose a value on the interval $x > 6$ and see if this value makes the original inequality true.

$x = 8$

Step 6.3.2

Replace $x$ with $8$ in the original inequality.

${((8) - 2)}^{2} (24 - 4 \cdot 8) > 0$

Step 6.3.3

The left side $- 288$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 6.4

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

$x < 2$ True

$2 < x < 6$ True

$x > 6$ False

$x < 2$ True

$2 < x < 6$ True

$x > 6$ False

Step 7

The solution consists of all of the true intervals.

$x < 2$ or $2 < x < 6$

Step 8

Convert the inequality to interval notation.

$(- \infty, 2) \cup (2, 6)$

Step 9