Solve the Inequality for x 2x^2+5x-50<=3x+10

$2 x^{2} + 5 x - 50 \leq 3 x + 10$

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

Subtract $3 x$ from both sides of the inequality.

$2 x^{2} + 5 x - 50 - 3 x \leq 10$

Subtract $3 x$ from $5 x$ .

$2 x^{2} + 2 x - 50 \leq 10$

Convert the inequality to an equation.

$2 x^{2} + 2 x - 50 = 10$

Subtract $10$ from both sides of the equation.

$2 x^{2} + 2 x - 50 - 10 = 0$

Subtract $10$ from $- 50$ .

$2 x^{2} + 2 x - 60 = 0$

Factor the left side of the equation.

Factor $2$ out of $2 x^{2} + 2 x - 60$ .

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

$2 (x^{2}) + 2 x - 60 = 0$

Factor $2$ out of $2 x$ .

$2 (x^{2}) + 2 (x) - 60 = 0$

Factor $2$ out of $- 60$ .

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

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

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

Factor $2$ out of $2 (x^{2} + x) + 2 \cdot - 30$ .

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

Factor.

Factor $x^{2} + x - 30$ 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 $- 30$ and whose sum is $1$ .

$- 5, 6$

Write the factored form using these integers.

$2 ((x - 5) (x + 6)) = 0$

Remove unnecessary parentheses.

$2 (x - 5) (x + 6) = 0$

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

$x - 5 = 0$

$x + 6 = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

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

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $2 (x - 5) (x + 6) = 0$ true.

$x = 5, - 6$

Use each root to create test intervals.

$x < - 6$

$- 6 < x < 5$

$x > 5$

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.

$x = - 8$

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

$2 {(- 8)}^{2} + 5 (- 8) - 50 \leq 3 (- 8) + 10$

The left side $38$ is greater than the right side $- 14$ , which means that the given statement is false.

False

Test a value on the interval $- 6 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $- 6 < x < 5$ and see if this value makes the original inequality true.

$x = 0$

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

$2 {(0)}^{2} + 5 (0) - 50 \leq 3 (0) + 10$

The left side $- 50$ is less than the right side $10$ , which means that the given statement is always true.

True

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.

$2 {(8)}^{2} + 5 (8) - 50 \leq 3 (8) + 10$

The left side $118$ is greater than the right side $34$ , which means that the given statement is false.

False

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

$x < - 6$ False

$- 6 < x < 5$ True

$x > 5$ False

$x < - 6$ False

$- 6 < x < 5$ True

$x > 5$ False

The solution consists of all of the true intervals.

$- 6 \leq x \leq 5$

The result can be shown in multiple forms.

Inequality Form:

$- 6 \leq x \leq 5$

Interval Notation:

$[- 6, 5]$

Solve the Inequality for x 2x^2+5x-50<=3x+10

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