Pre-Algebra Examples

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

Step 1

Combine $x$ and $\frac{1}{3}$ .

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Combine $\frac{1}{2}$ and $x$ .

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

Step 2.3

Multiply $\frac{1}{2} (- \frac{3}{5})$ .

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Step 2.3.1

Multiply $\frac{1}{2}$ by $\frac{3}{5}$ .

$1 - \frac{x}{3} \geq \frac{x}{2} - \frac{3}{2 \cdot 5}$

Step 2.3.2

Multiply $2$ by $5$ .

$1 - \frac{x}{3} \geq \frac{x}{2} - \frac{3}{10}$

Step 3

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

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Step 3.1

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

$1 - \frac{x}{3} - \frac{x}{2} \geq - \frac{3}{10}$

Step 3.2

To write $- \frac{x}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$1 - \frac{x}{3} \cdot \frac{2}{2} - \frac{x}{2} \geq - \frac{3}{10}$

Step 3.3

To write $- \frac{x}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$1 - \frac{x}{3} \cdot \frac{2}{2} - \frac{x}{2} \cdot \frac{3}{3} \geq - \frac{3}{10}$

Step 3.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.1

Multiply $\frac{x}{3}$ by $\frac{2}{2}$ .

$1 - \frac{x \cdot 2}{3 \cdot 2} - \frac{x}{2} \cdot \frac{3}{3} \geq - \frac{3}{10}$

Step 3.4.2

Multiply $3$ by $2$ .

$1 - \frac{x \cdot 2}{6} - \frac{x}{2} \cdot \frac{3}{3} \geq - \frac{3}{10}$

Step 3.4.3

Multiply $\frac{x}{2}$ by $\frac{3}{3}$ .

$1 - \frac{x \cdot 2}{6} - \frac{x \cdot 3}{2 \cdot 3} \geq - \frac{3}{10}$

Step 3.4.4

Multiply $2$ by $3$ .

$1 - \frac{x \cdot 2}{6} - \frac{x \cdot 3}{6} \geq - \frac{3}{10}$

Step 3.5

Combine the numerators over the common denominator.

$1 + \frac{- x \cdot 2 - x \cdot 3}{6} \geq - \frac{3}{10}$

Step 3.6

Simplify each term.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Factor $x$ out of $- x \cdot 2 - x \cdot 3$ .

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Step 3.6.1.1.1

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

$1 + \frac{x (- 1 \cdot 2) - x \cdot 3}{6} \geq - \frac{3}{10}$

Step 3.6.1.1.2

Factor $x$ out of $- x \cdot 3$ .

$1 + \frac{x (- 1 \cdot 2) + x (- 1 \cdot 3)}{6} \geq - \frac{3}{10}$

Step 3.6.1.1.3

Factor $x$ out of $x (- 1 \cdot 2) + x (- 1 \cdot 3)$ .

$1 + \frac{x (- 1 \cdot 2 - 1 \cdot 3)}{6} \geq - \frac{3}{10}$

Step 3.6.1.2

Multiply $- 1$ by $2$ .

$1 + \frac{x (- 2 - 1 \cdot 3)}{6} \geq - \frac{3}{10}$

Step 3.6.1.3

Multiply $- 1$ by $3$ .

$1 + \frac{x (- 2 - 3)}{6} \geq - \frac{3}{10}$

Step 3.6.1.4

Subtract $3$ from $- 2$ .

$1 + \frac{x \cdot - 5}{6} \geq - \frac{3}{10}$

Step 3.6.2

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

$1 + \frac{- 5 \cdot x}{6} \geq - \frac{3}{10}$

Step 3.6.3

Move the negative in front of the fraction.

$1 - \frac{5 x}{6} \geq - \frac{3}{10}$

Step 4

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

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Step 4.1

Subtract $1$ from both sides of the inequality.

$- \frac{5 x}{6} \geq - \frac{3}{10} - 1$

Step 4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$- \frac{5 x}{6} \geq - \frac{3}{10} - 1 \cdot \frac{10}{10}$

Step 4.3

Combine $- 1$ and $\frac{10}{10}$ .

$- \frac{5 x}{6} \geq - \frac{3}{10} + \frac{- 1 \cdot 10}{10}$

Step 4.4

Combine the numerators over the common denominator.

$- \frac{5 x}{6} \geq \frac{- 3 - 1 \cdot 10}{10}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $10$ .

$- \frac{5 x}{6} \geq \frac{- 3 - 10}{10}$

Step 4.5.2

Subtract $10$ from $- 3$ .

$- \frac{5 x}{6} \geq \frac{- 13}{10}$

Step 4.6

Move the negative in front of the fraction.

$- \frac{5 x}{6} \geq - \frac{13}{10}$

Step 5

Divide each term in $- \frac{5 x}{6} \geq - \frac{13}{10}$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- \frac{5 x}{6} \geq - \frac{13}{10}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{5 x}{6}}{- 1} \leq \frac{- \frac{13}{10}}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{5 x}{6}}{1} \leq \frac{- \frac{13}{10}}{- 1}$

Step 5.2.2

Divide $\frac{5 x}{6}$ by $1$ .

$\frac{5 x}{6} \leq \frac{- \frac{13}{10}}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Dividing two negative values results in a positive value.

$\frac{5 x}{6} \leq \frac{\frac{13}{10}}{1}$

Step 5.3.2

Divide $\frac{13}{10}$ by $1$ .

$\frac{5 x}{6} \leq \frac{13}{10}$

Step 6

Multiply both sides by $6$ .

$\frac{5 x}{6} \cdot 6 \leq \frac{13}{10} \cdot 6$

Step 7

Simplify.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Cancel the common factor of $6$ .

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Step 7.1.1.1

Cancel the common factor.

$\frac{5 x}{6} \cdot 6 \leq \frac{13}{10} \cdot 6$

Step 7.1.1.2

Rewrite the expression.

$5 x \leq \frac{13}{10} \cdot 6$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{13}{10} \cdot 6$ .

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Step 7.2.1.1

Cancel the common factor of $2$ .

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Step 7.2.1.1.1

Factor $2$ out of $10$ .

$5 x \leq \frac{13}{2 (5)} \cdot 6$

Step 7.2.1.1.2

Factor $2$ out of $6$ .

$5 x \leq \frac{13}{2 \cdot 5} \cdot (2 \cdot 3)$

Step 7.2.1.1.3

Cancel the common factor.

$5 x \leq \frac{13}{2 \cdot 5} \cdot (2 \cdot 3)$

Step 7.2.1.1.4

Rewrite the expression.

$5 x \leq \frac{13}{5} \cdot 3$

Step 7.2.1.2

Combine $\frac{13}{5}$ and $3$ .

$5 x \leq \frac{13 \cdot 3}{5}$

Step 7.2.1.3

Multiply $13$ by $3$ .

$5 x \leq \frac{39}{5}$

Step 8

Divide each term in $5 x \leq \frac{39}{5}$ by $5$ and simplify.

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Step 8.1

Divide each term in $5 x \leq \frac{39}{5}$ by $5$ .

$\frac{5 x}{5} \leq \frac{\frac{39}{5}}{5}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $5$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{5 x}{5} \leq \frac{\frac{39}{5}}{5}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{\frac{39}{5}}{5}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Multiply the numerator by the reciprocal of the denominator.

$x \leq \frac{39}{5} \cdot \frac{1}{5}$

Step 8.3.2

Multiply $\frac{39}{5} \cdot \frac{1}{5}$ .

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Step 8.3.2.1

Multiply $\frac{39}{5}$ by $\frac{1}{5}$ .

$x \leq \frac{39}{5 \cdot 5}$

Step 8.3.2.2

Multiply $5$ by $5$ .

$x \leq \frac{39}{25}$

Step 9

The result can be shown in multiple forms.

Inequality Form:

$x \leq \frac{39}{25}$

Interval Notation:

$(- \infty, \frac{39}{25}]$