Solve the Inequality for x x(x+4)<=-3

$x (x + 4) \leq - 3$

Step 1

Simplify $x (x + 4)$ .

Apply the distributive property.

$x \cdot x + x \cdot 4 \leq - 3$

Simplify the expression.

Multiply $x$ by $x$ .

$x^{2} + x \cdot 4 \leq - 3$

Move $4$ to the left of $x$ .

$x^{2} + 4 x \leq - 3$

Step 2

Convert the inequality to an equation.

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

Step 3

Add $3$ to both sides of the equation.

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

Step 4

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

$1, 3$

Write the factored form using these integers.

$(x + 1) (x + 3) = 0$

Step 5

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

$x + 1 = 0$

$x + 3 = 0$

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

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 8

The final solution is all the values that make $(x + 1) (x + 3) = 0$ true.

$x = - 1, - 3$

Step 9

Use each root to create test intervals.

$x < - 3$

$- 3 < x < - 1$

$x > - 1$

Step 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 $x < - 3$ to see if it makes the inequality true.

Choose a value on the interval $x < - 3$ and see if this value makes the original inequality true.

$x = - 6$

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

$(- 6) ((- 6) + 4) \leq - 3$

The left side $12$ is greater than the right side $- 3$ , which means that the given statement is false.

False

Test a value on the interval $- 3 < x < - 1$ to see if it makes the inequality true.

Choose a value on the interval $- 3 < x < - 1$ and see if this value makes the original inequality true.

$x = - 2$

Replace $x$ with $- 2$ in the original inequality.

$(- 2) ((- 2) + 4) \leq - 3$

The left side $- 4$ is less than the right side $- 3$ , which means that the given statement is always true.

True

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.

$x = 0$

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

$(0) ((0) + 4) \leq - 3$

The left side $0$ is greater than the right side $- 3$ , which means that the given statement is false.

False

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

$x < - 3$ False

$- 3 < x < - 1$ True

$x > - 1$ False

$x < - 3$ False

$- 3 < x < - 1$ True

$x > - 1$ False

Step 11

The solution consists of all of the true intervals.

$- 3 \leq x \leq - 1$

Step 12

The result can be shown in multiple forms.

Inequality Form:

$- 3 \leq x \leq - 1$

Interval Notation:

$[- 3, - 1]$

Step 13

Solve the Inequality for x x(x+4)<=-3

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