Pre-Algebra Examples

$| 3 x - 1 | + 2 > 5 x$

Step 1

Write $| 3 x - 1 | + 2 > 5 x$ 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.

$3 x - 1 \geq 0$

Step 1.2

Solve the inequality.

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Step 1.2.1

Add $1$ to both sides of the inequality.

$3 x \geq 1$

Step 1.2.2

Divide each term in $3 x \geq 1$ by $3$ and simplify.

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Step 1.2.2.1

Divide each term in $3 x \geq 1$ by $3$ .

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

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.3

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

$3 x - 1 + 2 > 5 x$

Step 1.4

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

$3 x - 1 < 0$

Step 1.5

Solve the inequality.

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Step 1.5.1

Add $1$ to both sides of the inequality.

$3 x < 1$

Step 1.5.2

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

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Step 1.5.2.1

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

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

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $3$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

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

Step 1.5.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.6

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

$- (3 x - 1) + 2 > 5 x$

Step 1.7

Write as a piecewise.

${\begin{cases} 3 x - 1 + 2 > 5 x & x \geq \frac{1}{3} \\ - (3 x - 1) + 2 > 5 x & x < \frac{1}{3} \end{cases}$

Step 1.8

Add $- 1$ and $2$ .

${\begin{cases} 3 x + 1 > 5 x & x \geq \frac{1}{3} \\ - (3 x - 1) + 2 > 5 x & x < \frac{1}{3} \end{cases}$

Step 1.9

Simplify $- (3 x - 1) + 2 > 5 x$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} 3 x + 1 > 5 x & x \geq \frac{1}{3} \\ - (3 x) - - 1 + 2 > 5 x & x < \frac{1}{3} \end{cases}$

Step 1.9.1.2

Multiply $3$ by $- 1$ .

${\begin{cases} 3 x + 1 > 5 x & x \geq \frac{1}{3} \\ - 3 x - - 1 + 2 > 5 x & x < \frac{1}{3} \end{cases}$

Step 1.9.1.3

Multiply $- 1$ by $- 1$ .

${\begin{cases} 3 x + 1 > 5 x & x \geq \frac{1}{3} \\ - 3 x + 1 + 2 > 5 x & x < \frac{1}{3} \end{cases}$

Step 1.9.2

Add $1$ and $2$ .

${\begin{cases} 3 x + 1 > 5 x & x \geq \frac{1}{3} \\ - 3 x + 3 > 5 x & x < \frac{1}{3} \end{cases}$

Step 2

Solve $3 x + 1 > 5 x$ when $x \geq \frac{1}{3}$ .

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Step 2.1

Solve $3 x + 1 > 5 x$ for $x$ .

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Step 2.1.1

Move all terms containing $x$ to the left side of the inequality.

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Step 2.1.1.1

Subtract $5 x$ from both sides of the inequality.

$3 x + 1 - 5 x > 0$

Step 2.1.1.2

Subtract $5 x$ from $3 x$ .

$- 2 x + 1 > 0$

Step 2.1.2

Subtract $1$ from both sides of the inequality.

$- 2 x > - 1$

Step 2.1.3

Divide each term in $- 2 x > - 1$ by $- 2$ and simplify.

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Step 2.1.3.1

Divide each term in $- 2 x > - 1$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} < \frac{- 1}{- 2}$

Step 2.1.3.2

Simplify the left side.

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Step 2.1.3.2.1

Cancel the common factor of $- 2$ .

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Step 2.1.3.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} < \frac{- 1}{- 2}$

Step 2.1.3.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 1}{- 2}$

Step 2.1.3.3

Simplify the right side.

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Step 2.1.3.3.1

Dividing two negative values results in a positive value.

$x < \frac{1}{2}$

Step 2.2

Find the intersection of $x < \frac{1}{2}$ and $x \geq \frac{1}{3}$ .

$\frac{1}{3} \leq x < \frac{1}{2}$

Step 3

Solve $- 3 x + 3 > 5 x$ when $x < \frac{1}{3}$ .

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Step 3.1

Solve $- 3 x + 3 > 5 x$ for $x$ .

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Step 3.1.1

Move all terms containing $x$ to the left side of the inequality.

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Step 3.1.1.1

Subtract $5 x$ from both sides of the inequality.

$- 3 x + 3 - 5 x > 0$

Step 3.1.1.2

Subtract $5 x$ from $- 3 x$ .

$- 8 x + 3 > 0$

Step 3.1.2

Subtract $3$ from both sides of the inequality.

$- 8 x > - 3$

Step 3.1.3

Divide each term in $- 8 x > - 3$ by $- 8$ and simplify.

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Step 3.1.3.1

Divide each term in $- 8 x > - 3$ by $- 8$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 8 x}{- 8} < \frac{- 3}{- 8}$

Step 3.1.3.2

Simplify the left side.

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Step 3.1.3.2.1

Cancel the common factor of $- 8$ .

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Step 3.1.3.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} < \frac{- 3}{- 8}$

Step 3.1.3.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 3}{- 8}$

Step 3.1.3.3

Simplify the right side.

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Step 3.1.3.3.1

Dividing two negative values results in a positive value.

$x < \frac{3}{8}$

Step 3.2

Find the intersection of $x < \frac{3}{8}$ and $x < \frac{1}{3}$ .

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

Step 4

Find the union of the solutions.

$x < \frac{1}{2}$

Step 5