Solve the Inequality for x 3x^2+45x+159<-3

$3 x^{2} + 45 x + 159 < - 3$

Convert the inequality to an equation.

$3 x^{2} + 45 x + 159 = - 3$

Add $3$ to both sides of the equation.

$3 x^{2} + 45 x + 159 + 3 = 0$

Add $159$ and $3$ .

$3 x^{2} + 45 x + 162 = 0$

Factor the left side of the equation.

Factor $3$ out of $3 x^{2} + 45 x + 162$ .

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) + 45 x + 162 = 0$

Factor $3$ out of $45 x$ .

$3 (x^{2}) + 3 (15 x) + 162 = 0$

Factor $3$ out of $162$ .

$3 x^{2} + 3 (15 x) + 3 \cdot 54 = 0$

Factor $3$ out of $3 x^{2} + 3 (15 x)$ .

$3 (x^{2} + 15 x) + 3 \cdot 54 = 0$

Factor $3$ out of $3 (x^{2} + 15 x) + 3 \cdot 54$ .

$3 (x^{2} + 15 x + 54) = 0$

Factor.

Factor $x^{2} + 15 x + 54$ 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 $54$ and whose sum is $15$ .

$6, 9$

Write the factored form using these integers.

$3 ((x + 6) (x + 9)) = 0$

Remove unnecessary parentheses.

$3 (x + 6) (x + 9) = 0$

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

$x + 6 = 0$

$x + 9 = 0$

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.

$x = - 6$

Set $x + 9$ equal to $0$ and solve for $x$ .

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Subtract $9$ from both sides of the equation.

$x = - 9$

The final solution is all the values that make $3 (x + 6) (x + 9) = 0$ true.

$x = - 6, - 9$

Use each root to create test intervals.

$x < - 9$

$- 9 < x < - 6$

$x > - 6$

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 $x < - 9$ to see if it makes the inequality true.

Choose a value on the interval $x < - 9$ and see if this value makes the original inequality true.

$x = - 12$

Replace $x$ with $- 12$ in the original inequality.

$3 {(- 12)}^{2} + 45 (- 12) + 159 < - 3$

The left side $51$ is not less than the right side $- 3$ , which means that the given statement is false.

False

Test a value on the interval $- 9 < x < - 6$ to see if it makes the inequality true.

Choose a value on the interval $- 9 < x < - 6$ and see if this value makes the original inequality true.

$x = - 8$

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

$3 {(- 8)}^{2} + 45 (- 8) + 159 < - 3$

The left side $- 9$ is less than the right side $- 3$ , which means that the given statement is always true.

True

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.

$x = 0$

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

$3 {(0)}^{2} + 45 (0) + 159 < - 3$

The left side $159$ is not less 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.

$x < - 9$ False

$- 9 < x < - 6$ True

$x > - 6$ False

$x < - 9$ False

$- 9 < x < - 6$ True

$x > - 6$ False

The solution consists of all of the true intervals.

$- 9 < x < - 6$

The result can be shown in multiple forms.

Inequality Form:

$- 9 < x < - 6$

Interval Notation:

$(- 9, - 6)$

Solve the Inequality for x 3x^2+45x+159<-3

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