Solve the Inequality for x 2x^2-28x+89>-7

$2 x^{2} - 28 x + 89 > - 7$

Step 1

Convert the inequality to an equation.

$2 x^{2} - 28 x + 89 = - 7$

Step 2

Add $7$ to both sides of the equation.

$2 x^{2} - 28 x + 89 + 7 = 0$

Step 3

Add $89$ and $7$ .

$2 x^{2} - 28 x + 96 = 0$

Step 4

Factor the left side of the equation.

Factor $2$ out of $2 x^{2} - 28 x + 96$ .

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) - 28 x + 96 = 0$

Factor $2$ out of $- 28 x$ .

$2 (x^{2}) + 2 (- 14 x) + 96 = 0$

Factor $2$ out of $96$ .

$2 x^{2} + 2 (- 14 x) + 2 \cdot 48 = 0$

Factor $2$ out of $2 x^{2} + 2 (- 14 x)$ .

$2 (x^{2} - 14 x) + 2 \cdot 48 = 0$

Factor $2$ out of $2 (x^{2} - 14 x) + 2 \cdot 48$ .

$2 (x^{2} - 14 x + 48) = 0$

Factor.

Factor $x^{2} - 14 x + 48$ 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 $48$ and whose sum is $- 14$ .

$- 8, - 6$

Write the factored form using these integers.

$2 ((x - 8) (x - 6)) = 0$

Remove unnecessary parentheses.

$2 (x - 8) (x - 6) = 0$

Step 5

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

$x - 8 = 0$

$x - 6 = 0$

Step 6

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

Step 7

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 8

The final solution is all the values that make $2 (x - 8) (x - 6) = 0$ true.

$x = 8, 6$

Step 9

Use each root to create test intervals.

$x < 6$

$6 < x < 8$

$x > 8$

Step 10

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.

$2 {(0)}^{2} - 28 \cdot 0 + 89 > - 7$

The left side $89$ is greater than the right side $- 7$ , which means that the given statement is always true.

True

Test a value on the interval $6 < x < 8$ to see if it makes the inequality true.

Choose a value on the interval $6 < x < 8$ and see if this value makes the original inequality true.

$x = 7$

Replace $x$ with $7$ in the original inequality.

$2 {(7)}^{2} - 28 \cdot 7 + 89 > - 7$

The left side $- 9$ is not greater than the right side $- 7$ , which means that the given statement is false.

False

Test a value on the interval $x > 8$ to see if it makes the inequality true.

Choose a value on the interval $x > 8$ and see if this value makes the original inequality true.

$x = 10$

Replace $x$ with $10$ in the original inequality.

$2 {(10)}^{2} - 28 \cdot 10 + 89 > - 7$

The left side $9$ is greater than the right side $- 7$ , which means that the given statement is always true.

True

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

$x < 6$ True

$6 < x < 8$ False

$x > 8$ True

$x < 6$ True

$6 < x < 8$ False

$x > 8$ True

Step 11

The solution consists of all of the true intervals.

$x < 6$ or $x > 8$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$x < 6 or x > 8$

Interval Notation:

$(- \infty, 6) \cup (8, \infty)$

Step 13

Solve the Inequality for x 2x^2-28x+89>-7

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