Precalculus Examples

$(2 x + 1) (3 x - 4) > 0$

Step 1

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

$2 x + 1 = 0$

$3 x - 4 = 0$

Step 2

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 2.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 2.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 2.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 2.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

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

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Move the negative in front of the fraction.

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

Step 3

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

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Step 3.1

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

$3 x - 4 = 0$

Step 3.2

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

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Step 3.2.1

Add $4$ to both sides of the equation.

$3 x = 4$

Step 3.2.2

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

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Step 3.2.2.1

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

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

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 4

The final solution is all the values that make $(2 x + 1) (3 x - 4) > 0$ true.

$x = - \frac{1}{2}, \frac{4}{3}$

Step 5

Use each root to create test intervals.

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

$- \frac{1}{2} < x < \frac{4}{3}$

$x > \frac{4}{3}$

Step 6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 6.1

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

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Step 6.1.1

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

$x = - 3$

Step 6.1.2

Replace $x$ with $- 3$ in the original inequality.

$(2 (- 3) + 1) (3 (- 3) - 4) > 0$

Step 6.1.3

The left side $65$ 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 $- \frac{1}{2} < x < \frac{4}{3}$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $- \frac{1}{2} < x < \frac{4}{3}$ and see if this value makes the original inequality true.

$x = 0$

Step 6.2.2

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

$(2 (0) + 1) (3 (0) - 4) > 0$

Step 6.2.3

The left side $- 4$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 6.3

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

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Step 6.3.1

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

$x = 4$

Step 6.3.2

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

$(2 (4) + 1) (3 (4) - 4) > 0$

Step 6.3.3

The left side $72$ 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.

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

$- \frac{1}{2} < x < \frac{4}{3}$ False

$x > \frac{4}{3}$ True

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

$- \frac{1}{2} < x < \frac{4}{3}$ False

$x > \frac{4}{3}$ True

Step 7

The solution consists of all of the true intervals.

$x < - \frac{1}{2}$ or $x > \frac{4}{3}$

Step 8

Convert the inequality to interval notation.

$(- \infty, - \frac{1}{2}) \cup (\frac{4}{3}, \infty)$

Step 9