Algebra Examples

$x | x | \leq 3$

Step 1

Write $x | x | \leq 3$ as a piecewise.

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Step 1.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 1.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \cdot x \leq 3$

Step 1.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 1.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$x (- x) \leq 3$

Step 1.5

Write as a piecewise.

${\begin{cases} x \cdot x \leq 3 & x \geq 0 \\ x (- x) \leq 3 & x < 0 \end{cases}$

Step 1.6

Multiply $x$ by $x$ .

${\begin{cases} x^{2} \leq 3 & x \geq 0 \\ x (- x) \leq 3 & x < 0 \end{cases}$

Step 1.7

Simplify $x (- x) \leq 3$ .

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Step 1.7.1

Rewrite using the commutative property of multiplication.

${\begin{cases} x^{2} \leq 3 & x \geq 0 \\ - x \cdot x \leq 3 & x < 0 \end{cases}$

Step 1.7.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.7.2.1

Move $x$ .

${\begin{cases} x^{2} \leq 3 & x \geq 0 \\ - (x \cdot x) \leq 3 & x < 0 \end{cases}$

Step 1.7.2.2

Multiply $x$ by $x$ .

${\begin{cases} x^{2} \leq 3 & x \geq 0 \\ - x^{2} \leq 3 & x < 0 \end{cases}$

Step 2

Solve $x^{2} \leq 3$ when $x \geq 0$ .

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Step 2.1

Solve $x^{2} \leq 3$ for $x$ .

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Step 2.1.1

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \leq \sqrt{3}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Pull terms out from under the radical.

$| x | \leq \sqrt{3}$

Step 2.1.3

Write $| x | \leq \sqrt{3}$ as a piecewise.

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Step 2.1.3.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 2.1.3.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq \sqrt{3}$

Step 2.1.3.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 2.1.3.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq \sqrt{3}$

Step 2.1.3.5

Write as a piecewise.

${\begin{cases} x \leq \sqrt{3} & x \geq 0 \\ - x \leq \sqrt{3} & x < 0 \end{cases}$

Step 2.1.4

Find the intersection of $x \leq \sqrt{3}$ and $x \geq 0$ .

$0 \leq x \leq \sqrt{3}$

Step 2.1.5

Solve $- x \leq \sqrt{3}$ when $x < 0$ .

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Step 2.1.5.1

Divide each term in $- x \leq \sqrt{3}$ by $- 1$ and simplify.

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Step 2.1.5.1.1

Divide each term in $- x \leq \sqrt{3}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{\sqrt{3}}{- 1}$

Step 2.1.5.1.2

Simplify the left side.

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Step 2.1.5.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{\sqrt{3}}{- 1}$

Step 2.1.5.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{\sqrt{3}}{- 1}$

Step 2.1.5.1.3

Simplify the right side.

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Step 2.1.5.1.3.1

Move the negative one from the denominator of $\frac{\sqrt{3}}{- 1}$ .

$x \geq - 1 \cdot \sqrt{3}$

Step 2.1.5.1.3.2

Rewrite $- 1 \cdot \sqrt{3}$ as $- \sqrt{3}$ .

$x \geq - \sqrt{3}$

Step 2.1.5.2

Find the intersection of $x \geq - \sqrt{3}$ and $x < 0$ .

$- \sqrt{3} \leq x < 0$

Step 2.1.6

Find the union of the solutions.

$- \sqrt{3} \leq x \leq \sqrt{3}$

Step 2.2

Find the intersection of $- \sqrt{3} \leq x \leq \sqrt{3}$ and $x \geq 0$ .

$0 \leq x \leq \sqrt{3}$

Step 3

Solve $- x^{2} \leq 3$ when $x < 0$ .

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Step 3.1

Solve $- x^{2} \leq 3$ for $x$ .

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Step 3.1.1

Divide each term in $- x^{2} \leq 3$ by $- 1$ and simplify.

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Step 3.1.1.1

Divide each term in $- x^{2} \leq 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \geq \frac{3}{- 1}$

Step 3.1.1.2

Simplify the left side.

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Step 3.1.1.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \geq \frac{3}{- 1}$

Step 3.1.1.2.2

Divide $x^{2}$ by $1$ .

$x^{2} \geq \frac{3}{- 1}$

Step 3.1.1.3

Simplify the right side.

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Step 3.1.1.3.1

Divide $3$ by $- 1$ .

$x^{2} \geq - 3$

Step 3.1.2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 3.2

Find the intersection.

$x < 0$

Step 4

Find the union of the solutions.

$x \leq \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x \leq \sqrt{3}$

Interval Notation:

$(- \infty, \sqrt{3}]$

Step 6