Algebra Examples

$1 \leq | x | \leq 4$

Step 1

Find the $x$ values that make $1 \leq | x |$ true.

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Step 1.1

Rewrite so $x$ is on the left side of the inequality.

$| x | \geq 1$

Step 1.2

Write $| x | \geq 1$ as a piecewise.

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Step 1.2.1

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

$x \geq 0$

Step 1.2.2

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

$x \geq 1$

Step 1.2.3

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

$x < 0$

Step 1.2.4

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

$- x \geq 1$

Step 1.2.5

Write as a piecewise.

${\begin{cases} x \geq 1 & x \geq 0 \\ - x \geq 1 & x < 0 \end{cases}$

Step 1.3

Find the intersection of $x \geq 1$ and $x \geq 0$ .

$x \geq 1$

Step 1.4

Divide each term in $- x \geq 1$ by $- 1$ and simplify.

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Step 1.4.1

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

$\frac{- x}{- 1} \leq \frac{1}{- 1}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{1}{- 1}$

Step 1.4.2.2

Divide $x$ by $1$ .

$x \leq \frac{1}{- 1}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Divide $1$ by $- 1$ .

$x \leq - 1$

Step 1.5

Find the union of the solutions.

$x \leq - 1$ or $x \geq 1$

Step 2

Find the $x$ values that make $| x | \leq 4$ true.

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Step 2.1

Write $| x | \leq 4$ as a piecewise.

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Step 2.1.1

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

$x \geq 0$

Step 2.1.2

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

$x \leq 4$

Step 2.1.3

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

$x < 0$

Step 2.1.4

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

$- x \leq 4$

Step 2.1.5

Write as a piecewise.

${\begin{cases} x \leq 4 & x \geq 0 \\ - x \leq 4 & x < 0 \end{cases}$

Step 2.2

Find the intersection of $x \leq 4$ and $x \geq 0$ .

$0 \leq x \leq 4$

Step 2.3

Solve $- x \leq 4$ when $x < 0$ .

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Step 2.3.1

Divide each term in $- x \leq 4$ by $- 1$ and simplify.

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Step 2.3.1.1

Divide each term in $- x \leq 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{4}{- 1}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{4}{- 1}$

Step 2.3.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{4}{- 1}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $4$ by $- 1$ .

$x \geq - 4$

Step 2.3.2

Find the intersection of $x \geq - 4$ and $x < 0$ .

$- 4 \leq x < 0$

Step 2.4

Find the union of the solutions.

$- 4 \leq x \leq 4$

Step 3

The solution is the intersection of the intervals.

$x \leq - 1$ or $x \geq 1$

$- 4 \leq x \leq 4$

Step 4

Find the intersection.

$- 4 \leq x \leq - 1$ or $1 \leq x \leq 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 4 \leq x \leq - 1 or 1 \leq x \leq 4$

Interval Notation:

$[- 4, - 1] \cup [1, 4]$

Step 6