Algebra Examples

$9 x \leq 12 x^{2}$

Step 1

Subtract $12 x^{2}$ from both sides of the inequality.

$9 x - 12 x^{2} \leq 0$

Step 2

Convert the inequality to an equation.

$9 x - 12 x^{2} = 0$

Step 3

Factor $3 x$ out of $9 x - 12 x^{2}$ .

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Step 3.1

Factor $3 x$ out of $9 x$ .

$3 x (3) - 12 x^{2} = 0$

Step 3.2

Factor $3 x$ out of $- 12 x^{2}$ .

$3 x (3) + 3 x (- 4 x) = 0$

Step 3.3

Factor $3 x$ out of $3 x (3) + 3 x (- 4 x)$ .

$3 x (3 - 4 x) = 0$

Step 4

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

$x = 0$

$3 - 4 x = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

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

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Step 6.1

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

$3 - 4 x = 0$

Step 6.2

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

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Step 6.2.1

Subtract $3$ from both sides of the equation.

$- 4 x = - 3$

Step 6.2.2

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

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Step 6.2.2.1

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

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

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $- 4$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 7

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

$x = 0, \frac{3}{4}$

Step 8

Use each root to create test intervals.

$x < 0$

$0 < x < \frac{3}{4}$

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

Step 9

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 9.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 9.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 9.1.2

Replace $x$ with $- 2$ in the original inequality.

$9 (- 2) \leq 12 {(- 2)}^{2}$

Step 9.1.3

The left side $- 18$ is less than the right side $48$ , which means that the given statement is always true.

True

Step 9.2

Test a value on the interval $0 < x < \frac{3}{4}$ to see if it makes the inequality true.

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Step 9.2.1

Choose a value on the interval $0 < x < \frac{3}{4}$ and see if this value makes the original inequality true.

$x = 0.38$

Step 9.2.2

Replace $x$ with $0.38$ in the original inequality.

$9 (0.38) \leq 12 {(0.38)}^{2}$

Step 9.2.3

The left side $3.42$ is greater than the right side $1.7328$ , which means that the given statement is false.

False

Step 9.3

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

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Step 9.3.1

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

$x = 3$

Step 9.3.2

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

$9 (3) \leq 12 {(3)}^{2}$

Step 9.3.3

The left side $27$ is less than the right side $108$ , which means that the given statement is always true.

True

Step 9.4

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

$x < 0$ True

$0 < x < \frac{3}{4}$ False

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

$x < 0$ True

$0 < x < \frac{3}{4}$ False

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

Step 10

The solution consists of all of the true intervals.

$x \leq 0$ or $x \geq \frac{3}{4}$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x \leq 0 or x \geq \frac{3}{4}$

Interval Notation:

$(- \infty, 0] \cup [\frac{3}{4}, \infty)$

Step 12