Algebra Examples

$| 3 x | + 5 < 4$

Step 1

Write $| 3 x | + 5 < 4$ 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.

$3 x \geq 0$

Step 1.2

Divide each term in $3 x \geq 0$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x \geq 0$ by $3$ .

$\frac{3 x}{3} \geq \frac{0}{3}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} \geq \frac{0}{3}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{3}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $3$ .

$x \geq 0$

Step 1.3

In the piece where $3 x$ is non-negative, remove the absolute value.

$3 x + 5 < 4$

Step 1.4

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

$3 x < 0$

Step 1.5

Divide each term in $3 x < 0$ by $3$ and simplify.

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Step 1.5.1

Divide each term in $3 x < 0$ by $3$ .

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

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Cancel the common factor of $3$ .

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Step 1.5.2.1.1

Cancel the common factor.

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

Step 1.5.2.1.2

Divide $x$ by $1$ .

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

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Divide $0$ by $3$ .

$x < 0$

Step 1.6

In the piece where $3 x$ is negative, remove the absolute value and multiply by $- 1$ .

$- 3 x + 5 < 4$

Step 1.7

Write as a piecewise.

${\begin{cases} 3 x + 5 < 4 & x \geq 0 \\ - 3 x + 5 < 4 & x < 0 \end{cases}$

Step 2

Solve $3 x + 5 < 4$ when $x \geq 0$ .

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Step 2.1

Solve $3 x + 5 < 4$ for $x$ .

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Step 2.1.1

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

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Step 2.1.1.1

Subtract $5$ from both sides of the inequality.

$3 x < 4 - 5$

Step 2.1.1.2

Subtract $5$ from $4$ .

$3 x < - 1$

Step 2.1.2

Divide each term in $3 x < - 1$ by $3$ and simplify.

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Step 2.1.2.1

Divide each term in $3 x < - 1$ by $3$ .

$\frac{3 x}{3} < \frac{- 1}{3}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $3$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} < \frac{- 1}{3}$

Step 2.1.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 1}{3}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Move the negative in front of the fraction.

$x < - \frac{1}{3}$

Step 2.2

Find the intersection of $x < - \frac{1}{3}$ and $x \geq 0$ .

No solution

Step 3

Solve $- 3 x + 5 < 4$ when $x < 0$ .

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Step 3.1

Solve $- 3 x + 5 < 4$ 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 $5$ from both sides of the inequality.

$- 3 x < 4 - 5$

Step 3.1.1.2

Subtract $5$ from $4$ .

$- 3 x < - 1$

Step 3.1.2

Divide each term in $- 3 x < - 1$ by $- 3$ and simplify.

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Step 3.1.2.1

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

$\frac{- 3 x}{- 3} > \frac{- 1}{- 3}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Cancel the common factor of $- 3$ .

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Step 3.1.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} > \frac{- 1}{- 3}$

Step 3.1.2.2.1.2

Divide $x$ by $1$ .

$x > \frac{- 1}{- 3}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Dividing two negative values results in a positive value.

$x > \frac{1}{3}$

Step 3.2

Find the intersection of $x > \frac{1}{3}$ and $x < 0$ .

No solution

Step 4

Find the union of the solutions.

No solution