Precalculus Examples

∣ ∣ x^{2} + 4 x + 4 ∣ ∣

$| x^{2} + 4 x + 4 |$

Step 1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

x^{2} + 4 x + 4 \geq 0

$x^{2} + 4 x + 4 \geq 0$

Step 2

Solve the inequality.

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Step 2.1

Convert the inequality to an equation.

x^{2} + 4 x + 4 = 0

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

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

x^{2} + 4 x + 2^{2} = 0

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

Step 2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

4 x = 2 \cdot x \cdot 2

$4 x = 2 \cdot x \cdot 2$

Step 2.2.3

Rewrite the polynomial.

x^{2} + 2 \cdot x \cdot 2 + 2^{2} = 0

$x^{2} + 2 \cdot x \cdot 2 + 2^{2} = 0$

Step 2.2.4

Factor using the perfect square trinomial rule

a^{2} + 2 a b + b^{2} = {(a + b)}^{2}

$a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where

a = x

$a = x$ and

b = 2

$b = 2$ .

{(x + 2)}^{2} = 0

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

{(x + 2)}^{2} = 0

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

Step 2.3

Set the

x + 2

$x + 2$ equal to

0

$0$ .

x + 2 = 0

$x + 2 = 0$

Step 2.4

Subtract

2

$2$ from both sides of the equation.

x = - 2

$x = - 2$

Step 2.5

Use each root to create test intervals.

x < - 2

$x < - 2$

x > - 2

$x > - 2$

Step 2.6

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 2.6.1

Test a value on the interval

x < - 2

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

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Step 2.6.1.1

Choose a value on the interval

x < - 2

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

x = - 4

$x = - 4$

Step 2.6.1.2

Replace

x

$x$ with

- 4

$- 4$ in the original inequality.

{(- 4)}^{2} + 4 (- 4) + 4 \geq 0

${(- 4)}^{2} + 4 (- 4) + 4 \geq 0$

Step 2.6.1.3

The left side

4

$4$ is greater than the right side

0

$0$ , which means that the given statement is always true.

True

Step 2.6.2

Test a value on the interval

x > - 2

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

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Step 2.6.2.1

Choose a value on the interval

x > - 2

$x > - 2$ and see if this value makes the original inequality true.

x = 0

$x = 0$

Step 2.6.2.2

Replace

x

$x$ with

0

$0$ in the original inequality.

{(0)}^{2} + 4 (0) + 4 \geq 0

${(0)}^{2} + 4 (0) + 4 \geq 0$

Step 2.6.2.3

The left side

4

$4$ is greater than the right side

0

$0$ , which means that the given statement is always true.

True

Step 2.6.3

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

x < - 2

$x < - 2$ True

x > - 2

$x > - 2$ True

x < - 2

$x < - 2$ True

x > - 2

$x > - 2$ True

Step 2.7

The solution consists of all of the true intervals.

x \leq - 2

$x \leq - 2$ or

x \geq - 2

$x \geq - 2$

Step 2.8

Combine the intervals.

All real numbers

Step 3

Since

x^{2} + 4 x + 4

$x^{2} + 4 x + 4$ is never negative, the absolute value can be removed.

x^{2} + 4 x + 4

$x^{2} + 4 x + 4$