Solve the Inequality for x -2x^2+15x+150<9x+10

$- 2 x^{2} + 15 x + 150 < 9 x + 10$

Move all terms containing $x$ to the left side of the inequality.

Subtract $9 x$ from both sides of the inequality.

$- 2 x^{2} + 15 x + 150 - 9 x < 10$

Subtract $9 x$ from $15 x$ .

$- 2 x^{2} + 6 x + 150 < 10$

Convert the inequality to an equation.

$- 2 x^{2} + 6 x + 150 = 10$

Subtract $10$ from both sides of the equation.

$- 2 x^{2} + 6 x + 150 - 10 = 0$

Subtract $10$ from $150$ .

$- 2 x^{2} + 6 x + 140 = 0$

Factor the left side of the equation.

Factor $- 2$ out of $- 2 x^{2} + 6 x + 140$ .

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

$- 2 x^{2} + 6 x + 140 = 0$

Factor $- 2$ out of $6 x$ .

$- 2 x^{2} - 2 (- 3 x) + 140 = 0$

Factor $- 2$ out of $140$ .

$- 2 x^{2} - 2 (- 3 x) - 2 \cdot - 70 = 0$

Factor $- 2$ out of $- 2 (x^{2}) - 2 (- 3 x)$ .

$- 2 (x^{2} - 3 x) - 2 \cdot - 70 = 0$

Factor $- 2$ out of $- 2 (x^{2} - 3 x) - 2 (- 70)$ .

$- 2 (x^{2} - 3 x - 70) = 0$

Factor.

Factor $x^{2} - 3 x - 70$ 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 $- 70$ and whose sum is $- 3$ .

$- 10, 7$

Write the factored form using these integers.

$- 2 ((x - 10) (x + 7)) = 0$

Remove unnecessary parentheses.

$- 2 (x - 10) (x + 7) = 0$

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

$x - 10 = 0$

$x + 7 = 0$

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.

$x = 10$

Set $x + 7$ equal to $0$ and solve for $x$ .

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

The final solution is all the values that make $- 2 (x - 10) (x + 7) = 0$ true.

$x = 10, - 7$

Use each root to create test intervals.

$x < - 7$

$- 7 < x < 10$

$x > 10$

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 $x < - 7$ to see if it makes the inequality true.

Choose a value on the interval $x < - 7$ and see if this value makes the original inequality true.

$x = - 10$

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

$- 2 {(- 10)}^{2} + 15 (- 10) + 150 < 9 (- 10) + 10$

The left side $- 200$ is less than the right side $- 80$ , which means that the given statement is always true.

True

Test a value on the interval $- 7 < x < 10$ to see if it makes the inequality true.

Choose a value on the interval $- 7 < x < 10$ and see if this value makes the original inequality true.

$x = 0$

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

$- 2 {(0)}^{2} + 15 (0) + 150 < 9 (0) + 10$

The left side $150$ is not less than the right side $10$ , 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.

$- 2 {(12)}^{2} + 15 (12) + 150 < 9 (12) + 10$

The left side $42$ is less than the right side $118$ , which means that the given statement is always true.

True

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

$x < - 7$ True

$- 7 < x < 10$ False

$x > 10$ True

$x < - 7$ True

$- 7 < x < 10$ False

$x > 10$ True

The solution consists of all of the true intervals.

$x < - 7$ or $x > 10$

The result can be shown in multiple forms.

Inequality Form:

$x < - 7 or x > 10$

Interval Notation:

$(- \infty, - 7) \cup (10, \infty)$

Solve the Inequality for x -2x^2+15x+150<9x+10

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