Trigonometry Examples

$23 + | x | > 19$

Step 1

Write $23 + | x | > 19$ 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.

$23 + x > 19$

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 $- 1$ .

$23 - x > 19$

Step 1.5

Write as a piecewise.

${\begin{cases} 23 + x > 19 & x \geq 0 \\ 23 - x > 19 & x < 0 \end{cases}$

Step 2

Solve $23 + x > 19$ when $x \geq 0$ .

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Step 2.1

Move all terms not containing $x$ to the right side of the inequality.

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Step 2.1.1

Subtract $23$ from both sides of the inequality.

$x > 19 - 23$

Step 2.1.2

Subtract $23$ from $19$ .

$x > - 4$

Step 2.2

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

$x \geq 0$

Step 3

Solve $23 - x > 19$ when $x < 0$ .

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Step 3.1

Solve $23 - x > 19$ for $x$ .

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Step 3.1.1

Move all terms not containing $x$ to the right side of the inequality.

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Step 3.1.1.1

Subtract $23$ from both sides of the inequality.

$- x > 19 - 23$

Step 3.1.1.2

Subtract $23$ from $19$ .

$- x > - 4$

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.1.2.2.2

Divide $x$ by $1$ .

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

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $- 4$ by $- 1$ .

$x < 4$

Step 3.2

Find the intersection of $x < 4$ and $x < 0$ .

$x < 0$

Step 4

Find the union of the solutions for any value of $x$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Step 6