Solve the Inequality for x 2x^2+19x-28>9x

$2 x^{2} + 19 x - 28 > 9 x$

Step 1

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

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

$2 x^{2} + 19 x - 28 - 9 x > 0$

Subtract $9 x$ from $19 x$ .

$2 x^{2} + 10 x - 28 > 0$

Step 2

Convert the inequality to an equation.

$2 x^{2} + 10 x - 28 = 0$

Step 3

Factor the left side of the equation.

Factor $2$ out of $2 x^{2} + 10 x - 28$ .

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

$2 (x^{2}) + 10 x - 28 = 0$

Factor $2$ out of $10 x$ .

$2 (x^{2}) + 2 (5 x) - 28 = 0$

Factor $2$ out of $- 28$ .

$2 x^{2} + 2 (5 x) + 2 \cdot - 14 = 0$

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

$2 (x^{2} + 5 x) + 2 \cdot - 14 = 0$

Factor $2$ out of $2 (x^{2} + 5 x) + 2 \cdot - 14$ .

$2 (x^{2} + 5 x - 14) = 0$

Factor.

Factor $x^{2} + 5 x - 14$ 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 $- 14$ and whose sum is $5$ .

$- 2, 7$

Write the factored form using these integers.

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

Remove unnecessary parentheses.

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

Step 4

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

$x - 2 = 0$

$x + 7 = 0$

Step 5

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 6

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.

$x = - 7$

Step 7

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

$x = 2, - 7$

Step 8

Use each root to create test intervals.

$x < - 7$

$- 7 < x < 2$

$x > 2$

Step 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 $x < - 7$ 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.

$2 {(- 10)}^{2} + 19 (- 10) - 28 > 9 (- 10)$

The left side $- 18$ is greater than the right side $- 90$ , which means that the given statement is always true.

True

Test a value on the interval $- 7 < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- 7 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$2 {(0)}^{2} + 19 (0) - 28 > 9 (0)$

The left side $- 28$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > 2$ to see if it makes the inequality true.

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

$2 {(4)}^{2} + 19 (4) - 28 > 9 (4)$

The left side $80$ is greater than the right side $36$ , which means that the given statement is always true.

True

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

$x < - 7$ True

$- 7 < x < 2$ False

$x > 2$ True

$x < - 7$ True

$- 7 < x < 2$ False

$x > 2$ True

Step 10

The solution consists of all of the true intervals.

$x < - 7$ or $x > 2$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x < - 7 or x > 2$

Interval Notation:

$(- \infty, - 7) \cup (2, \infty)$

Step 12

Solve the Inequality for x 2x^2+19x-28>9x

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