Solve the Inequality for x 3x^2-5x-8<-3x

$3 x^{2} - 5 x - 8 < - 3 x$

Step 1

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

Add $3 x$ to both sides of the inequality.

$3 x^{2} - 5 x - 8 + 3 x < 0$

Add $- 5 x$ and $3 x$ .

$3 x^{2} - 2 x - 8 < 0$

Step 2

Convert the inequality to an equation.

$3 x^{2} - 2 x - 8 = 0$

Step 3

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 8 = - 24$ and whose sum is $b = - 2$ .

Factor $- 2$ out of $- 2 x$ .

$3 x^{2} - 2 x - 8 = 0$

Rewrite $- 2$ as $4$ plus $- 6$

$3 x^{2} + (4 - 6) x - 8 = 0$

Apply the distributive property.

$3 x^{2} + 4 x - 6 x - 8 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(3 x^{2} + 4 x) - 6 x - 8 = 0$

Factor out the greatest common factor (GCF) from each group.

$x (3 x + 4) - 2 (3 x + 4) = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x + 4$ .

$(3 x + 4) (x - 2) = 0$

Step 4

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

$3 x + 4 = 0$

$x - 2 = 0$

Step 5

Set $3 x + 4$ equal to $0$ and solve for $x$ .

Set $3 x + 4$ equal to $0$ .

$3 x + 4 = 0$

Solve $3 x + 4 = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$3 x = - 4$

Divide each term in $3 x = - 4$ by $3$ and simplify.

Divide each term in $3 x = - 4$ by $3$ .

$\frac{3 x}{3} = \frac{- 4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 4}{3}$

Divide $x$ by $1$ .

$x = \frac{- 4}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{4}{3}$

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 $(3 x + 4) (x - 2) = 0$ true.

$x = - \frac{4}{3}, 2$

Step 8

Use each root to create test intervals.

$x < - \frac{4}{3}$

$- \frac{4}{3} < x < 2$

$x > 2$

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 $x < - \frac{4}{3}$ to see if it makes the inequality true.

Choose a value on the interval $x < - \frac{4}{3}$ and see if this value makes the original inequality true.

$x = - 4$

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

$3 {(- 4)}^{2} - 5 \cdot - 4 - 8 < - 3 \cdot - 4$

The left side $60$ is not less than the right side $12$ , which means that the given statement is false.

False

Test a value on the interval $- \frac{4}{3} < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{4}{3} < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$3 {(0)}^{2} - 5 \cdot 0 - 8 < - 3 \cdot 0$

The left side $- 8$ is less than the right side $0$ , which means that the given statement is always true.

True

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.

$3 {(4)}^{2} - 5 \cdot 4 - 8 < - 3 \cdot 4$

The left side $20$ is not less than the right side $- 12$ , which means that the given statement is false.

False

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

$x < - \frac{4}{3}$ False

$- \frac{4}{3} < x < 2$ True

$x > 2$ False

$x < - \frac{4}{3}$ False

$- \frac{4}{3} < x < 2$ True

$x > 2$ False

Step 10

The solution consists of all of the true intervals.

$- \frac{4}{3} < x < 2$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$- \frac{4}{3} < x < 2$

Interval Notation:

$(- \frac{4}{3}, 2)$

Step 12

Solve the Inequality for x 3x^2-5x-8<-3x

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