Precalculus Examples

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

Step 1

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

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

Step 2

Solve the inequality.

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Step 2.1

Convert the inequality to an equation.

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

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

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

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

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.

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

Step 2.2.3

Rewrite the polynomial.

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

Step 2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 2$ .

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

Step 2.3

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

$x + 2 = 0$

Step 2.4

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.5

Use each root to create test intervals.

$x < - 2$

$x > - 2$

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

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Step 2.6.1.1

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

$x = - 4$

Step 2.6.1.2

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

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

Step 2.6.1.3

The left side $4$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.6.2

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

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Step 2.6.2.1

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

$x = 0$

Step 2.6.2.2

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

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

Step 2.6.2.3

The left side $4$ is greater than the right side $0$ , 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.

$x < - 2$ True

$x > - 2$ True

$x < - 2$ True

$x > - 2$ True

Step 2.7

The solution consists of all of the true intervals.

$x \leq - 2$ or $x \geq - 2$

Step 2.8

Combine the intervals.

All real numbers

Step 3

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

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