Algebra Examples

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

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 $x$ .

Tap for more steps...

Step 2.1

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

$2 x + 3 = 0$

Step 2.2

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

Tap for more steps...

Step 2.2.1

Subtract $3$ from both sides of the equation.

$2 x = - 3$

Step 2.2.2

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

Tap for more steps...

Step 2.2.2.1

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

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

Step 2.2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.2.3.1

Move the negative in front of the fraction.

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

Step 3

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

Tap for more steps...

Step 3.1

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

$x - 1 = 0$

Step 3.2

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.

Tap for more steps...

Step 6.1

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

Tap for more steps...

Step 6.1.1

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

$x = - 4$

Step 6.1.2

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

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

Step 6.1.3

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

True

Step 6.2

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

Tap for more steps...

Step 6.2.1

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

$x = 0$

Step 6.2.2

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

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

Step 6.2.3

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

False

Step 6.3

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

Tap for more steps...

Step 6.3.1

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

$x = 4$

Step 6.3.2

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

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

Step 6.3.3

The left side $33$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.4

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