Pre-Algebra Examples

$\frac{2 (1 - 3 x)}{5} \leq 1 + \frac{x}{2}$

Step 1

Multiply both sides by $5$ .

$\frac{2 (1 - 3 x)}{5} \cdot 5 \leq (1 + \frac{x}{2}) \cdot 5$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{2 (1 - 3 x)}{5} \cdot 5$ .

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Step 2.1.1.1

Cancel the common factor of $5$ .

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Step 2.1.1.1.1

Cancel the common factor.

$\frac{2 (1 - 3 x)}{5} \cdot 5 \leq (1 + \frac{x}{2}) \cdot 5$

Step 2.1.1.1.2

Rewrite the expression.

$2 (1 - 3 x) \leq (1 + \frac{x}{2}) \cdot 5$

Step 2.1.1.2

Apply the distributive property.

$2 \cdot 1 + 2 (- 3 x) \leq (1 + \frac{x}{2}) \cdot 5$

Step 2.1.1.3

Simplify the expression.

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Step 2.1.1.3.1

Multiply $2$ by $1$ .

$2 + 2 (- 3 x) \leq (1 + \frac{x}{2}) \cdot 5$

Step 2.1.1.3.2

Multiply $- 3$ by $2$ .

$2 - 6 x \leq (1 + \frac{x}{2}) \cdot 5$

Step 2.1.1.3.3

Reorder $2$ and $- 6 x$ .

$- 6 x + 2 \leq (1 + \frac{x}{2}) \cdot 5$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(1 + \frac{x}{2}) \cdot 5$ .

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Step 2.2.1.1

Simplify terms.

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Step 2.2.1.1.1

Apply the distributive property.

$- 6 x + 2 \leq 1 \cdot 5 + \frac{x}{2} \cdot 5$

Step 2.2.1.1.2

Multiply $5$ by $1$ .

$- 6 x + 2 \leq 5 + \frac{x}{2} \cdot 5$

Step 2.2.1.1.3

Combine $\frac{x}{2}$ and $5$ .

$- 6 x + 2 \leq 5 + \frac{x \cdot 5}{2}$

Step 2.2.1.2

Move $5$ to the left of $x$ .

$- 6 x + 2 \leq 5 + \frac{5 x}{2}$

Step 2.2.1.3

Reorder $5$ and $\frac{5 x}{2}$ .

$- 6 x + 2 \leq \frac{5 x}{2} + 5$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $\frac{5 x}{2}$ from both sides of the inequality.

$- 6 x + 2 - \frac{5 x}{2} \leq 5$

Step 3.1.2

To write $- 6 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 6 x \cdot \frac{2}{2} - \frac{5 x}{2} + 2 \leq 5$

Step 3.1.3

Combine $- 6 x$ and $\frac{2}{2}$ .

$\frac{- 6 x \cdot 2}{2} - \frac{5 x}{2} + 2 \leq 5$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{- 6 x \cdot 2 - 5 x}{2} + 2 \leq 5$

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

Simplify the numerator.

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Step 3.1.5.1.1

Factor $x$ out of $- 6 x \cdot 2 - 5 x$ .

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Step 3.1.5.1.1.1

Factor $x$ out of $- 6 x \cdot 2$ .

$\frac{x (- 6 \cdot 2) - 5 x}{2} + 2 \leq 5$

Step 3.1.5.1.1.2

Factor $x$ out of $- 5 x$ .

$\frac{x (- 6 \cdot 2) + x \cdot - 5}{2} + 2 \leq 5$

Step 3.1.5.1.1.3

Factor $x$ out of $x (- 6 \cdot 2) + x \cdot - 5$ .

$\frac{x (- 6 \cdot 2 - 5)}{2} + 2 \leq 5$

Step 3.1.5.1.2

Multiply $- 6$ by $2$ .

$\frac{x (- 12 - 5)}{2} + 2 \leq 5$

Step 3.1.5.1.3

Subtract $5$ from $- 12$ .

$\frac{x \cdot - 17}{2} + 2 \leq 5$

Step 3.1.5.2

Move $- 17$ to the left of $x$ .

$\frac{- 17 \cdot x}{2} + 2 \leq 5$

Step 3.1.5.3

Move the negative in front of the fraction.

$- \frac{17 x}{2} + 2 \leq 5$

Step 3.2

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

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Step 3.2.1

Subtract $2$ from both sides of the inequality.

$- \frac{17 x}{2} \leq 5 - 2$

Step 3.2.2

Subtract $2$ from $5$ .

$- \frac{17 x}{2} \leq 3$

Step 3.3

Divide each term in $- \frac{17 x}{2} \leq 3$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- \frac{17 x}{2} \leq 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{17 x}{2}}{- 1} \geq \frac{3}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{17 x}{2}}{1} \geq \frac{3}{- 1}$

Step 3.3.2.2

Divide $\frac{17 x}{2}$ by $1$ .

$\frac{17 x}{2} \geq \frac{3}{- 1}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $3$ by $- 1$ .

$\frac{17 x}{2} \geq - 3$

Step 3.4

Multiply both sides by $2$ .

$\frac{17 x}{2} \cdot 2 \geq - 3 \cdot 2$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Cancel the common factor of $2$ .

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Step 3.5.1.1.1

Cancel the common factor.

$\frac{17 x}{2} \cdot 2 \geq - 3 \cdot 2$

Step 3.5.1.1.2

Rewrite the expression.

$17 x \geq - 3 \cdot 2$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Multiply $- 3$ by $2$ .

$17 x \geq - 6$

Step 3.6

Divide each term in $17 x \geq - 6$ by $17$ and simplify.

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Step 3.6.1

Divide each term in $17 x \geq - 6$ by $17$ .

$\frac{17 x}{17} \geq \frac{- 6}{17}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $17$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{17 x}{17} \geq \frac{- 6}{17}$

Step 3.6.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 6}{17}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Move the negative in front of the fraction.

$x \geq - \frac{6}{17}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \geq - \frac{6}{17}$

Interval Notation:

$[- \frac{6}{17}, \infty)$