Solve the Inequality for x (3x+5)(4x-8)>0

$(3 x + 5) (4 x - 8) > 0$

Step 1

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

$3 x + 5 = 0$

$4 x - 8 = 0$

Step 2

Set $3 x + 5$ equal to $0$ and solve for $x$ .

Set $3 x + 5$ equal to $0$ .

$3 x + 5 = 0$

Solve $3 x + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$3 x = - 5$

Divide each term in $3 x = - 5$ by $3$ and simplify.

Divide each term in $3 x = - 5$ by $3$ .

$\frac{3 x}{3} = \frac{- 5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 5}{3}$

Divide $x$ by $1$ .

$x = \frac{- 5}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5}{3}$

Step 3

Set $4 x - 8$ equal to $0$ and solve for $x$ .

Set $4 x - 8$ equal to $0$ .

$4 x - 8 = 0$

Solve $4 x - 8 = 0$ for $x$ .

Add $8$ to both sides of the equation.

$4 x = 8$

Divide each term in $4 x = 8$ by $4$ and simplify.

Divide each term in $4 x = 8$ by $4$ .

$\frac{4 x}{4} = \frac{8}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{8}{4}$

Divide $x$ by $1$ .

$x = \frac{8}{4}$

Simplify the right side.

Divide $8$ by $4$ .

$x = 2$

Step 4

The final solution is all the values that make $(3 x + 5) (4 x - 8) > 0$ true.

$x = - \frac{5}{3}, 2$

Step 5

Use each root to create test intervals.

$x < - \frac{5}{3}$

$- \frac{5}{3} < x < 2$

$x > 2$

Step 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 $x < - \frac{5}{3}$ to see if it makes the inequality true.

Choose a value on the interval $x < - \frac{5}{3}$ and see if this value makes the original inequality true.

$x = - 4$

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

$(3 (- 4) + 5) (4 (- 4) - 8) > 0$

The left side $168$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- \frac{5}{3} < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{5}{3} < x < 2$ and see if this value makes the original inequality true.

$x = 0$

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

$(3 (0) + 5) (4 (0) - 8) > 0$

The left side $- 40$ is not greater than the right side $0$ , which means that the given statement is false.

False

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.

$(3 (4) + 5) (4 (4) - 8) > 0$

The left side $136$ is greater than the right side $0$ , which means that the given statement is always true.

True

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

$x < - \frac{5}{3}$ True

$- \frac{5}{3} < x < 2$ False

$x > 2$ True

$x < - \frac{5}{3}$ True

$- \frac{5}{3} < x < 2$ False

$x > 2$ True

Step 7

The solution consists of all of the true intervals.

$x < - \frac{5}{3}$ or $x > 2$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{5}{3} or x > 2$

Interval Notation:

$(- \infty, - \frac{5}{3}) \cup (2, \infty)$

Step 9

Solve the Inequality for x (3x+5)(4x-8)>0

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