Solve the Inequality for x 3x+4>x^2

$3 x + 4 > x^{2}$

Step 1

Subtract $x^{2}$ from both sides of the inequality.

$3 x + 4 - x^{2} > 0$

Step 2

Convert the inequality to an equation.

$3 x + 4 - x^{2} = 0$

Step 3

Factor the left side of the equation.

Factor $- 1$ out of $3 x + 4 - x^{2}$ .

Reorder the expression.

Move $4$ .

$3 x - x^{2} + 4 = 0$

Reorder $3 x$ and $- x^{2}$ .

$- x^{2} + 3 x + 4 = 0$

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 3 x + 4 = 0$

Factor $- 1$ out of $3 x$ .

$- (x^{2}) - (- 3 x) + 4 = 0$

Rewrite $4$ as $- 1 (- 4)$ .

$- (x^{2}) - (- 3 x) - 1 \cdot - 4 = 0$

Factor $- 1$ out of $- (x^{2}) - (- 3 x)$ .

$- (x^{2} - 3 x) - 1 \cdot - 4 = 0$

Factor $- 1$ out of $- (x^{2} - 3 x) - 1 (- 4)$ .

$- (x^{2} - 3 x - 4) = 0$

Factor.

Factor $x^{2} - 3 x - 4$ 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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Write the factored form using these integers.

$- ((x - 4) (x + 1)) = 0$

Remove unnecessary parentheses.

$- (x - 4) (x + 1) = 0$

Step 4

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

$x - 4 = 0$

$x + 1 = 0$

Step 5

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Step 6

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 7

The final solution is all the values that make $- (x - 4) (x + 1) = 0$ true.

$x = 4, - 1$

Step 8

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 4$

$x > 4$

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 $x < - 1$ to see if it makes the inequality true.

Choose a value on the interval $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

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

$3 (- 4) + 4 > {(- 4)}^{2}$

The left side $- 8$ is not greater than the right side $16$ , which means that the given statement is false.

False

Test a value on the interval $- 1 < x < 4$ to see if it makes the inequality true.

Choose a value on the interval $- 1 < x < 4$ and see if this value makes the original inequality true.

$x = 0$

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

$3 (0) + 4 > {(0)}^{2}$

The left side $4$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 4$ to see if it makes the inequality true.

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Replace $x$ with $6$ in the original inequality.

$3 (6) + 4 > {(6)}^{2}$

The left side $22$ is not greater than the right side $36$ , which means that the given statement is false.

False

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

$x < - 1$ False

$- 1 < x < 4$ True

$x > 4$ False

$x < - 1$ False

$- 1 < x < 4$ True

$x > 4$ False

Step 10

The solution consists of all of the true intervals.

$- 1 < x < 4$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$- 1 < x < 4$

Interval Notation:

$(- 1, 4)$

Step 12

Solve the Inequality for x 3x+4>x^2

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