Calculus Examples

$\frac{x^{2} - 3 x - 4}{x^{2} - 4 x + 5} < 0$

Step 1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x^{2} - 3 x - 4 = 0$

$x^{2} - 4 x + 5 = 0$

Step 2

Factor $x^{2} - 3 x - 4$ using the AC method.

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Step 2.1

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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 2.2

Write the factored form using these integers.

$(x - 4) (x + 1) = 0$

Step 3

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

$x - 4 = 0$

$x + 1 = 0$

Step 4

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

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Step 4.1

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

$x - 4 = 0$

Step 4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 5

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

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Step 5.1

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

$x + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 6

The final solution is all the values that make $(x - 4) (x + 1) = 0$ true.

$x = 4, - 1$

Step 7

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 8

Substitute the values $a = 1$ , $b = - 4$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 5)}}{2 \cdot 1}$

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $5$ .

$x = \frac{4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Step 9.1.3

Subtract $20$ from $16$ .

$x = \frac{4 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 9.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 9.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 9.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 9.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 9.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{4 \pm i \cdot 2}{2 \cdot 1}$

Step 9.1.9

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 i}{2}$

Step 9.3

Simplify $\frac{4 \pm 2 i}{2}$ .

$x = 2 \pm i$

Step 10

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 10.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 10.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Step 10.1.2.2

Multiply $- 4$ by $5$ .

$x = \frac{4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Step 10.1.3

Subtract $20$ from $16$ .

$x = \frac{4 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 10.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 10.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 10.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 10.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 10.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{4 \pm i \cdot 2}{2 \cdot 1}$

Step 10.1.9

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i}{2 \cdot 1}$

Step 10.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 i}{2}$

Step 10.3

Simplify $\frac{4 \pm 2 i}{2}$ .

$x = 2 \pm i$

Step 10.4

Change the $\pm$ to $+$ .

$x = 2 + i$

Step 11

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 11.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 11.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Step 11.1.2.2

Multiply $- 4$ by $5$ .

$x = \frac{4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Step 11.1.3

Subtract $20$ from $16$ .

$x = \frac{4 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 11.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 11.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 11.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 11.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 11.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{4 \pm i \cdot 2}{2 \cdot 1}$

Step 11.1.9

Move $2$ to the left of $i$ .

$x = \frac{4 \pm 2 i}{2 \cdot 1}$

Step 11.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 i}{2}$

Step 11.3

Simplify $\frac{4 \pm 2 i}{2}$ .

$x = 2 \pm i$

Step 11.4

Change the $\pm$ to $-$ .

$x = 2 - i$

Step 12

The final answer is the combination of both solutions.

$x = 2 + i, 2 - i$

Step 13

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 4, - 1$

$x = 2 + i, 2 - i$

Step 14

Consolidate the solutions.

$x = 4, - 1$

Step 15

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 4$

$x > 4$

Step 16

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 16.1

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

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Step 16.1.1

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

$x = - 4$

Step 16.1.2

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

$\frac{{(- 4)}^{2} - 3 \cdot - 4 - 4}{{(- 4)}^{2} - 4 \cdot - 4 + 5} < 0$

Step 16.1.3

The left side $0.\overline{648}$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 16.2

Test a value on the interval $- 1 < x < 4$ to see if it makes the inequality true.

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Step 16.2.1

Choose a value on the interval $- 1 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

Step 16.2.2

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

$\frac{{(0)}^{2} - 3 \cdot 0 - 4}{{(0)}^{2} - 4 \cdot 0 + 5} < 0$

Step 16.2.3

The left side $- 0.8$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 16.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 16.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 16.3.2

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

$\frac{{(6)}^{2} - 3 \cdot 6 - 4}{{(6)}^{2} - 4 \cdot 6 + 5} < 0$

Step 16.3.3

The left side $0.82352941$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 16.4

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

$x < - 1$ False

$- 1 < x < 4$ True

$x > 4$ False

$x < - 1$ False

$- 1 < x < 4$ True

$x > 4$ False

Step 17

The solution consists of all of the true intervals.

$- 1 < x < 4$

Step 18

Convert the inequality to interval notation.

$(- 1, 4)$

Step 19