Precalculus Examples

$| 7 x | < 3$

Step 1

Write $| 7 x | < 3$ 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.

$7 x \geq 0$

Step 1.2

Divide each term in $7 x \geq 0$ by $7$ and simplify.

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Step 1.2.1

Divide each term in $7 x \geq 0$ by $7$ .

$\frac{7 x}{7} \geq \frac{0}{7}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $7$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} \geq \frac{0}{7}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{7}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $7$ .

$x \geq 0$

Step 1.3

In the piece where $7 x$ is non-negative, remove the absolute value.

$7 x < 3$

Step 1.4

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

$7 x < 0$

Step 1.5

Divide each term in $7 x < 0$ by $7$ and simplify.

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Step 1.5.1

Divide each term in $7 x < 0$ by $7$ .

$\frac{7 x}{7} < \frac{0}{7}$

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Cancel the common factor of $7$ .

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Step 1.5.2.1.1

Cancel the common factor.

$\frac{7 x}{7} < \frac{0}{7}$

Step 1.5.2.1.2

Divide $x$ by $1$ .

$x < \frac{0}{7}$

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Divide $0$ by $7$ .

$x < 0$

Step 1.6

In the piece where $7 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- 7 x < 3$

Step 1.7

Write as a piecewise.

${\begin{cases} 7 x < 3 & x \geq 0 \\ - 7 x < 3 & x < 0 \end{cases}$

Step 2

Solve $7 x < 3$ when $x \geq 0$ .

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Step 2.1

Divide each term in $7 x < 3$ by $7$ and simplify.

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Step 2.1.1

Divide each term in $7 x < 3$ by $7$ .

$\frac{7 x}{7} < \frac{3}{7}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $7$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{7 x}{7} < \frac{3}{7}$

Step 2.1.2.1.2

Divide $x$ by $1$ .

$x < \frac{3}{7}$

Step 2.2

Find the intersection of $x < \frac{3}{7}$ and $x \geq 0$ .

$0 \leq x < \frac{3}{7}$

Step 3

Solve $- 7 x < 3$ when $x < 0$ .

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Step 3.1

Divide each term in $- 7 x < 3$ by $- 7$ and simplify.

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Step 3.1.1

Divide each term in $- 7 x < 3$ by $- 7$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 7 x}{- 7} > \frac{3}{- 7}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $- 7$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} > \frac{3}{- 7}$

Step 3.1.2.1.2

Divide $x$ by $1$ .

$x > \frac{3}{- 7}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Move the negative in front of the fraction.

$x > - \frac{3}{7}$

Step 3.2

Find the intersection of $x > - \frac{3}{7}$ and $x < 0$ .

$- \frac{3}{7} < x < 0$

Step 4

Find the union of the solutions.

$- \frac{3}{7} < x < \frac{3}{7}$

Step 5

Convert the inequality to interval notation.

$(- \frac{3}{7}, \frac{3}{7})$

Step 6