Finite Math Examples

$31 > 13 + 19 | x |$

Step 1

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

$13 + 19 | x | < 31$

Step 2

Write $13 + 19 | x | < 31$ as a piecewise.

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Step 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 2.2

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

$13 + 19 x < 31$

Step 2.3

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

$x < 0$

Step 2.4

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

$13 + 19 (- x) < 31$

Step 2.5

Write as a piecewise.

${\begin{cases} 13 + 19 x < 31 & x \geq 0 \\ 13 + 19 (- x) < 31 & x < 0 \end{cases}$

Step 2.6

Multiply $- 1$ by $19$ .

${\begin{cases} 13 + 19 x < 31 & x \geq 0 \\ 13 - 19 x < 31 & x < 0 \end{cases}$

Step 3

Solve $13 + 19 x < 31$ when $x \geq 0$ .

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Step 3.1

Solve $13 + 19 x < 31$ 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 $13$ from both sides of the inequality.

$19 x < 31 - 13$

Step 3.1.1.2

Subtract $13$ from $31$ .

$19 x < 18$

Step 3.1.2

Divide each term in $19 x < 18$ by $19$ and simplify.

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Step 3.1.2.1

Divide each term in $19 x < 18$ by $19$ .

$\frac{19 x}{19} < \frac{18}{19}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $19$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{19 x}{19} < \frac{18}{19}$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{18}{19}$

Step 3.2

Find the intersection of $x < \frac{18}{19}$ and $x \geq 0$ .

$0 \leq x < \frac{18}{19}$

Step 4

Solve $13 - 19 x < 31$ when $x < 0$ .

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Step 4.1

Solve $13 - 19 x < 31$ for $x$ .

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Step 4.1.1

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

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Step 4.1.1.1

Subtract $13$ from both sides of the inequality.

$- 19 x < 31 - 13$

Step 4.1.1.2

Subtract $13$ from $31$ .

$- 19 x < 18$

Step 4.1.2

Divide each term in $- 19 x < 18$ by $- 19$ and simplify.

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Step 4.1.2.1

Divide each term in $- 19 x < 18$ by $- 19$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 19 x}{- 19} > \frac{18}{- 19}$

Step 4.1.2.2

Simplify the left side.

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Step 4.1.2.2.1

Cancel the common factor of $- 19$ .

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Step 4.1.2.2.1.1

Cancel the common factor.

$\frac{- 19 x}{- 19} > \frac{18}{- 19}$

Step 4.1.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{18}{- 19}$

Step 4.1.2.3

Simplify the right side.

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Step 4.1.2.3.1

Move the negative in front of the fraction.

$x > - \frac{18}{19}$

Step 4.2

Find the intersection of $x > - \frac{18}{19}$ and $x < 0$ .

$- \frac{18}{19} < x < 0$

Step 5

Find the union of the solutions.

$- \frac{18}{19} < x < \frac{18}{19}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$- \frac{18}{19} < x < \frac{18}{19}$

Interval Notation:

$(- \frac{18}{19}, \frac{18}{19})$

Step 7