Solve the Inequality for x 2x^2-30x+104>=-4

$2 x^{2} - 30 x + 104 \geq - 4$

Convert the inequality to an equation.

$2 x^{2} - 30 x + 104 = - 4$

Add $4$ to both sides of the equation.

$2 x^{2} - 30 x + 104 + 4 = 0$

Add $104$ and $4$ .

$2 x^{2} - 30 x + 108 = 0$

Factor the left side of the equation.

Factor $2$ out of $2 x^{2} - 30 x + 108$ .

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

$2 (x^{2}) - 30 x + 108 = 0$

Factor $2$ out of $- 30 x$ .

$2 (x^{2}) + 2 (- 15 x) + 108 = 0$

Factor $2$ out of $108$ .

$2 x^{2} + 2 (- 15 x) + 2 \cdot 54 = 0$

Factor $2$ out of $2 x^{2} + 2 (- 15 x)$ .

$2 (x^{2} - 15 x) + 2 \cdot 54 = 0$

Factor $2$ out of $2 (x^{2} - 15 x) + 2 \cdot 54$ .

$2 (x^{2} - 15 x + 54) = 0$

Factor.

Factor $x^{2} - 15 x + 54$ 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 $54$ and whose sum is $- 15$ .

$- 9, - 6$

Write the factored form using these integers.

$2 ((x - 9) (x - 6)) = 0$

Remove unnecessary parentheses.

$2 (x - 9) (x - 6) = 0$

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

$x - 9 = 0$

$x - 6 = 0$

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

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.

$x = 6$

The final solution is all the values that make $2 (x - 9) (x - 6) = 0$ true.

$x = 9, 6$

Use each root to create test intervals.

$x < 6$

$6 < x < 9$

$x > 9$

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.

$2 {(0)}^{2} - 30 \cdot 0 + 104 \geq - 4$

The left side $104$ is greater than the right side $- 4$ , which means that the given statement is always true.

True

Test a value on the interval $6 < x < 9$ to see if it makes the inequality true.

Choose a value on the interval $6 < x < 9$ and see if this value makes the original inequality true.

$x = 8$

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

$2 {(8)}^{2} - 30 \cdot 8 + 104 \geq - 4$

The left side $- 8$ is less than the right side $- 4$ , which means that the given statement is false.

False

Test a value on the interval $x > 9$ to see if it makes the inequality true.

Choose a value on the interval $x > 9$ and see if this value makes the original inequality true.

$x = 12$

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

$2 {(12)}^{2} - 30 \cdot 12 + 104 \geq - 4$

The left side $32$ is greater than the right side $- 4$ , which means that the given statement is always true.

True

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

$x < 6$ True

$6 < x < 9$ False

$x > 9$ True

$x < 6$ True

$6 < x < 9$ False

$x > 9$ True

The solution consists of all of the true intervals.

$x \leq 6$ or $x \geq 9$

The result can be shown in multiple forms.

Inequality Form:

$x \leq 6 or x \geq 9$

Interval Notation:

$(- \infty, 6] \cup [9, \infty)$

Solve the Inequality for x 2x^2-30x+104>=-4

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