Solve the Inequality for x (x-1)(3x-4)>=0

$(x - 1) (3 x - 4) \geq 0$

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

$x - 1 = 0$

$3 x - 4 = 0$

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.

$x = 1$

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 $x$ .

Add $4$ to 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 $1$ .

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

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

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

Use each root to create test intervals.

$x < 1$

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

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

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 $x < 1$ 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.

$x = 0$

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

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

The left side $4$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $1 < x < \frac{4}{3}$ to see if it makes the inequality true.

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

$x = 1.17$

Replace $x$ with $1.17$ in the original inequality.

$((1.17) - 1) (3 (1.17) - 4) \geq 0$

The left side $- 0.0833$ is less than the right side $0$ , which means that the given statement is false.

False

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.

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

The left side $24$ 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.

$x < 1$ True

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

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

$x < 1$ True

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

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

The solution consists of all of the true intervals.

$x \leq 1$ or $x \geq \frac{4}{3}$

The result can be shown in multiple forms.

Inequality Form:

$x \leq 1 or x \geq \frac{4}{3}$

Interval Notation:

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

Solve the Inequality for x (x-1)(3x-4)>=0

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