Solve the Inequality for x (2x+3)(x-1)>=0

$(2 x + 3) (x - 1) \geq 0$

Step 1

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

$2 x + 3 = 0$

$x - 1 = 0$

Step 2

Set $2 x + 3$ equal to $0$ and solve for $x$ .

Set $2 x + 3$ equal to $0$ .

$2 x + 3 = 0$

Solve $2 x + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Divide each term in $2 x = - 3$ by $2$ and simplify.

Divide each term in $2 x = - 3$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 3

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Step 4

The final solution is all the values that make $(2 x + 3) (x - 1) \geq 0$ true.

$x = - \frac{3}{2}, 1$

Step 5

Use each root to create test intervals.

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

$- \frac{3}{2} < x < 1$

$x > 1$

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 $x < - \frac{3}{2}$ to see if it makes the inequality true.

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

$x = - 4$

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

$(2 (- 4) + 3) ((- 4) - 1) \geq 0$

The left side $25$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- \frac{3}{2} < x < 1$ to see if it makes the inequality true.

Choose a value on the interval $- \frac{3}{2} < x < 1$ and see if this value makes the original inequality true.

$x = 0$

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

$(2 (0) + 3) ((0) - 1) \geq 0$

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

False

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.

$(2 (4) + 3) ((4) - 1) \geq 0$

The left side $33$ 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.

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

$- \frac{3}{2} < x < 1$ False

$x > 1$ True

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

$- \frac{3}{2} < x < 1$ False

$x > 1$ True

Step 7

The solution consists of all of the true intervals.

$x \leq - \frac{3}{2}$ or $x \geq 1$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \frac{3}{2} or x \geq 1$

Interval Notation:

$(- \infty, - \frac{3}{2}] \cup [1, \infty)$

Step 9

Solve the Inequality for x (2x+3)(x-1)>=0

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