Solve the Inequality for x -4x^2+28x<40

$- 4 x^{2} + 28 x < 40$

Step 1

Convert the inequality to an equation.

$- 4 x^{2} + 28 x = 40$

Step 2

Subtract $40$ from both sides of the equation.

$- 4 x^{2} + 28 x - 40 = 0$

Step 3

Factor the left side of the equation.

Factor $- 4$ out of $- 4 x^{2} + 28 x - 40$ .

Factor $- 4$ out of $- 4 x^{2}$ .

$- 4 x^{2} + 28 x - 40 = 0$

Factor $- 4$ out of $28 x$ .

$- 4 x^{2} - 4 (- 7 x) - 40 = 0$

Factor $- 4$ out of $- 40$ .

$- 4 x^{2} - 4 (- 7 x) - 4 \cdot 10 = 0$

Factor $- 4$ out of $- 4 (x^{2}) - 4 (- 7 x)$ .

$- 4 (x^{2} - 7 x) - 4 \cdot 10 = 0$

Factor $- 4$ out of $- 4 (x^{2} - 7 x) - 4 (10)$ .

$- 4 (x^{2} - 7 x + 10) = 0$

Factor.

Factor $x^{2} - 7 x + 10$ 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 $10$ and whose sum is $- 7$ .

$- 5, - 2$

Write the factored form using these integers.

$- 4 ((x - 5) (x - 2)) = 0$

Remove unnecessary parentheses.

$- 4 (x - 5) (x - 2) = 0$

Step 4

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

$x - 5 = 0$

$x - 2 = 0$

Step 5

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.

$x = 5$

Step 6

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 7

The final solution is all the values that make $- 4 (x - 5) (x - 2) = 0$ true.

$x = 5, 2$

Step 8

Use each root to create test intervals.

$x < 2$

$2 < x < 5$

$x > 5$

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 $x < 2$ 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.

$x = 0$

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

$- 4 {(0)}^{2} + 28 (0) < 40$

The left side $0$ is less than the right side $40$ , which means that the given statement is always true.

True

Test a value on the interval $2 < x < 5$ to see if it makes the inequality true.

Choose a value on the interval $2 < x < 5$ and see if this value makes the original inequality true.

$x = 4$

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

$- 4 {(4)}^{2} + 28 (4) < 40$

The left side $48$ is not less than the right side $40$ , which means that the given statement is false.

False

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.

$- 4 {(8)}^{2} + 28 (8) < 40$

The left side $- 32$ is less than the right side $40$ , which means that the given statement is always true.

True

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

$x < 2$ True

$2 < x < 5$ False

$x > 5$ True

$x < 2$ True

$2 < x < 5$ False

$x > 5$ True

Step 10

The solution consists of all of the true intervals.

$x < 2$ or $x > 5$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x < 2 or x > 5$

Interval Notation:

$(- \infty, 2) \cup (5, \infty)$

Step 12

Solve the Inequality for x -4x^2+28x<40

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