Graph x^2+60>16x

$x^{2} + 60 > 16 x$

Step 1

Subtract $16 x$ from both sides of the inequality.

$x^{2} + 60 - 16 x > 0$

Step 2

Convert the inequality to an equation.

$x^{2} + 60 - 16 x = 0$

Step 3

Factor $x^{2} + 60 - 16 x$ using the AC method.

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 $60$ and whose sum is $- 16$ .

$- 10, - 6$

Write the factored form using these integers.

$(x - 10) (x - 6) = 0$

Step 4

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

$x - 10 = 0$

$x - 6 = 0$

Step 5

Set $x - 10$ equal to $0$ and solve for $x$ .

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Add $10$ to both sides of the equation.

$x = 10$

Step 6

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Step 7

The final solution is all the values that make $(x - 10) (x - 6) = 0$ true.

$x = 10, 6$

Step 8

Use each root to create test intervals.

$x < 6$

$6 < x < 10$

$x > 10$

Step 9

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 $x < 6$ to see if it makes the inequality true.

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

$x = 0$

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

${(0)}^{2} + 60 > 16 (0)$

The left side $60$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $6 < x < 10$ to see if it makes the inequality true.

Choose a value on the interval $6 < x < 10$ and see if this value makes the original inequality true.

$x = 8$

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

${(8)}^{2} + 60 > 16 (8)$

The left side $124$ is not greater than the right side $128$ , which means that the given statement is false.

False

Test a value on the interval $x > 10$ to see if it makes the inequality true.

Choose a value on the interval $x > 10$ and see if this value makes the original inequality true.

$x = 12$

Replace $x$ with $12$ in the original inequality.

${(12)}^{2} + 60 > 16 (12)$

The left side $204$ is greater than the right side $192$ , which means that the given statement is always true.

True

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

$x < 6$ True

$6 < x < 10$ False

$x > 10$ True

$x < 6$ True

$6 < x < 10$ False

$x > 10$ True

Step 10

The solution consists of all of the true intervals.

$x < 6$ or $x > 10$

Step 11

Graph x^2+60>16x

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