Solve the Inequality for x x^2-8x+16>0

$x^{2} - 8 x + 16 > 0$

Step 1

Convert the inequality to an equation.

$x^{2} - 8 x + 16 = 0$

Step 2

Factor using the perfect square rule.

Rewrite $16$ as $4^{2}$ .

$x^{2} - 8 x + 4^{2} = 0$

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

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 4 + 4^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

${(x - 4)}^{2} = 0$

Step 3

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

$x - 4 = 0$

Step 4

Add $4$ to both sides of the equation.

$x = 4$

Step 5

Use each root to create test intervals.

$x < 4$

$x > 4$

Step 6

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

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

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

$x = 0$

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

${(0)}^{2} - 8 \cdot 0 + 16 > 0$

The left side $16$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

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

$x = 6$

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

${(6)}^{2} - 8 \cdot 6 + 16 > 0$

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

True

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

$x < 4$ True

$x > 4$ True

$x < 4$ True

$x > 4$ True

Step 7

The solution consists of all of the true intervals.

$x < 4$ or $x > 4$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x < 4 or x > 4$

Interval Notation:

$(- \infty, 4) \cup (4, \infty)$

Step 9

Solve the Inequality for x x^2-8x+16>0

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