Linear Algebra Examples

$x - 3 \sqrt{3 x^{4} - 2 x^{3} - x - 1}$

Step 1

Set the radicand in $\sqrt{3 x^{4} - 2 x^{3} - x - 1}$ greater than or equal to $0$ to find where the expression is defined.

$3 x^{4} - 2 x^{3} - x - 1 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 0.51518557, 1.14400199$

Step 2.2

Use each root to create test intervals.

$x < - 0.51518557$

$- 0.51518557 < x < 1.14400199$

$x > 1.14400199$

Step 2.3

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 2.3.1

Test a value on the interval $x < - 0.51518557$ to see if it makes the inequality true.

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Step 2.3.1.1

Choose a value on the interval $x < - 0.51518557$ and see if this value makes the original inequality true.

$x = - 3$

Step 2.3.1.2

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

$3 {(- 3)}^{4} - 2 {(- 3)}^{3} - (- 3) - 1 \geq 0$

Step 2.3.1.3

The left side $299$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.3.2

Test a value on the interval $- 0.51518557 < x < 1.14400199$ to see if it makes the inequality true.

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Step 2.3.2.1

Choose a value on the interval $- 0.51518557 < x < 1.14400199$ and see if this value makes the original inequality true.

$x = 0$

Step 2.3.2.2

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

$3 {(0)}^{4} - 2 {(0)}^{3} - (0) - 1 \geq 0$

Step 2.3.2.3

The left side $- 1$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.3.3

Test a value on the interval $x > 1.14400199$ to see if it makes the inequality true.

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Step 2.3.3.1

Choose a value on the interval $x > 1.14400199$ and see if this value makes the original inequality true.

$x = 4$

Step 2.3.3.2

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

$3 {(4)}^{4} - 2 {(4)}^{3} - (4) - 1 \geq 0$

Step 2.3.3.3

The left side $635$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.3.4

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

$x < - 0.51518557$ True

$- 0.51518557 < x < 1.14400199$ False

$x > 1.14400199$ True

$x < - 0.51518557$ True

$- 0.51518557 < x < 1.14400199$ False

$x > 1.14400199$ True

Step 2.4

The solution consists of all of the true intervals.

$x \leq - 0.51518557$ or $x \geq 1.14400199$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 0.51518557] \cup [1.14400199, \infty)$

Set-Builder Notation:

${x | x \leq - 0.51518557, x \geq 1.14400199}$

Step 4