Solve the Inequality for x -2x^2+12x>18

$- 2 x^{2} + 12 x > 18$

Step 1

Convert the inequality to an equation.

$- 2 x^{2} + 12 x = 18$

Step 2

Subtract $18$ from both sides of the equation.

$- 2 x^{2} + 12 x - 18 = 0$

Step 3

Factor the left side of the equation.

Factor $- 2$ out of $- 2 x^{2} + 12 x - 18$ .

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

$- 2 x^{2} + 12 x - 18 = 0$

Factor $- 2$ out of $12 x$ .

$- 2 x^{2} - 2 (- 6 x) - 18 = 0$

Factor $- 2$ out of $- 18$ .

$- 2 x^{2} - 2 (- 6 x) - 2 \cdot 9 = 0$

Factor $- 2$ out of $- 2 (x^{2}) - 2 (- 6 x)$ .

$- 2 (x^{2} - 6 x) - 2 \cdot 9 = 0$

Factor $- 2$ out of $- 2 (x^{2} - 6 x) - 2 (9)$ .

$- 2 (x^{2} - 6 x + 9) = 0$

Factor using the perfect square rule.

Rewrite $9$ as $3^{2}$ .

$- 2 (x^{2} - 6 x + 3^{2}) = 0$

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

$6 x = 2 \cdot x \cdot 3$

Rewrite the polynomial.

$- 2 (x^{2} - 2 \cdot x \cdot 3 + 3^{2}) = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

$- 2 {(x - 3)}^{2} = 0$

Step 4

Divide each term in $- 2 {(x - 3)}^{2} = 0$ by $- 2$ and simplify.

Divide each term in $- 2 {(x - 3)}^{2} = 0$ by $- 2$ .

$\frac{- 2 {(x - 3)}^{2}}{- 2} = \frac{0}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 {(x - 3)}^{2}}{- 2} = \frac{0}{- 2}$

Divide ${(x - 3)}^{2}$ by $1$ .

${(x - 3)}^{2} = \frac{0}{- 2}$

Simplify the right side.

Divide $0$ by $- 2$ .

${(x - 3)}^{2} = 0$

Step 5

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 6

Add $3$ to both sides of the equation.

$x = 3$

Step 7

Use each root to create test intervals.

$x < 3$

$x > 3$

Step 8

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 $x < 3$ to see if it makes the inequality true.

Choose a value on the interval $x < 3$ and see if this value makes the original inequality true.

$x = 0$

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

$- 2 {(0)}^{2} + 12 (0) > 18$

The left side $0$ is not greater than the right side $18$ , which means that the given statement is false.

False

Test a value on the interval $x > 3$ to see if it makes the inequality true.

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

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

$- 2 {(6)}^{2} + 12 (6) > 18$

The left side $0$ is not greater than the right side $18$ , which means that the given statement is false.

False

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

$x < 3$ False

$x > 3$ False

$x < 3$ False

$x > 3$ False

Step 9

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Solve the Inequality for x -2x^2+12x>18

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