Solve the Inequality for x -2x^2+20x>=10x-100

$- 2 x^{2} + 20 x \geq 10 x - 100$

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

Subtract $10 x$ from both sides of the inequality.

$- 2 x^{2} + 20 x - 10 x \geq - 100$

Subtract $10 x$ from $20 x$ .

$- 2 x^{2} + 10 x \geq - 100$

Convert the inequality to an equation.

$- 2 x^{2} + 10 x = - 100$

Add $100$ to both sides of the equation.

$- 2 x^{2} + 10 x + 100 = 0$

Factor the left side of the equation.

Factor $- 2$ out of $- 2 x^{2} + 10 x + 100$ .

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

$- 2 x^{2} + 10 x + 100 = 0$

Factor $- 2$ out of $10 x$ .

$- 2 x^{2} - 2 (- 5 x) + 100 = 0$

Factor $- 2$ out of $100$ .

$- 2 x^{2} - 2 (- 5 x) - 2 \cdot - 50 = 0$

Factor $- 2$ out of $- 2 (x^{2}) - 2 (- 5 x)$ .

$- 2 (x^{2} - 5 x) - 2 \cdot - 50 = 0$

Factor $- 2$ out of $- 2 (x^{2} - 5 x) - 2 (- 50)$ .

$- 2 (x^{2} - 5 x - 50) = 0$

Factor.

Factor $x^{2} - 5 x - 50$ 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 $- 50$ and whose sum is $- 5$ .

$- 10, 5$

Write the factored form using these integers.

$- 2 ((x - 10) (x + 5)) = 0$

Remove unnecessary parentheses.

$- 2 (x - 10) (x + 5) = 0$

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

$x - 10 = 0$

$x + 5 = 0$

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 $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

The final solution is all the values that make $- 2 (x - 10) (x + 5) = 0$ true.

$x = 10, - 5$

Use each root to create test intervals.

$x < - 5$

$- 5 < x < 10$

$x > 10$

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 $x < - 5$ to see if it makes the inequality true.

Choose a value on the interval $x < - 5$ and see if this value makes the original inequality true.

$x = - 8$

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

$- 2 {(- 8)}^{2} + 20 (- 8) \geq 10 (- 8) - 100$

The left side $- 288$ is less than the right side $- 180$ , which means that the given statement is false.

False

Test a value on the interval $- 5 < x < 10$ to see if it makes the inequality true.

Choose a value on the interval $- 5 < x < 10$ and see if this value makes the original inequality true.

$x = 0$

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

$- 2 {(0)}^{2} + 20 (0) \geq 10 (0) - 100$

The left side $0$ is greater than the right side $- 100$ , which means that the given statement is always true.

True

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.

$- 2 {(12)}^{2} + 20 (12) \geq 10 (12) - 100$

The left side $- 48$ is less than the right side $20$ , which means that the given statement is false.

False

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

$x < - 5$ False

$- 5 < x < 10$ True

$x > 10$ False

$x < - 5$ False

$- 5 < x < 10$ True

$x > 10$ False

The solution consists of all of the true intervals.

$- 5 \leq x \leq 10$

The result can be shown in multiple forms.

Inequality Form:

$- 5 \leq x \leq 10$

Interval Notation:

$[- 5, 10]$

Solve the Inequality for x -2x^2+20x>=10x-100

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