Pre-Algebra Examples

$\frac{1}{5} x - \frac{2}{3} x - 2 = - \frac{2}{5} x$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{1}{5}$ and $x$ .

$\frac{x}{5} - \frac{2}{3} x - 2 = - \frac{2}{5} x$

Step 1.1.2

Combine $x$ and $\frac{2}{3}$ .

$\frac{x}{5} - \frac{x \cdot 2}{3} - 2 = - \frac{2}{5} x$

Step 1.1.3

Move $2$ to the left of $x$ .

$\frac{x}{5} - \frac{2 x}{3} - 2 = - \frac{2}{5} x$

Step 1.2

To write $\frac{x}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{5} \cdot \frac{3}{3} - \frac{2 x}{3} - 2 = - \frac{2}{5} x$

Step 1.3

To write $- \frac{2 x}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{x}{5} \cdot \frac{3}{3} - \frac{2 x}{3} \cdot \frac{5}{5} - 2 = - \frac{2}{5} x$

Step 1.4

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{x}{5}$ by $\frac{3}{3}$ .

$\frac{x \cdot 3}{5 \cdot 3} - \frac{2 x}{3} \cdot \frac{5}{5} - 2 = - \frac{2}{5} x$

Step 1.4.2

Multiply $5$ by $3$ .

$\frac{x \cdot 3}{15} - \frac{2 x}{3} \cdot \frac{5}{5} - 2 = - \frac{2}{5} x$

Step 1.4.3

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

$\frac{x \cdot 3}{15} - \frac{2 x \cdot 5}{3 \cdot 5} - 2 = - \frac{2}{5} x$

Step 1.4.4

Multiply $3$ by $5$ .

$\frac{x \cdot 3}{15} - \frac{2 x \cdot 5}{15} - 2 = - \frac{2}{5} x$

Step 1.5

Combine the numerators over the common denominator.

$\frac{x \cdot 3 - 2 x \cdot 5}{15} - 2 = - \frac{2}{5} x$

Step 1.6

Simplify each term.

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Factor $x$ out of $x \cdot 3 - 2 x \cdot 5$ .

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Step 1.6.1.1.1

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

$\frac{x (3) - 2 x \cdot 5}{15} - 2 = - \frac{2}{5} x$

Step 1.6.1.1.2

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

$\frac{x (3) + x (- 2 \cdot 5)}{15} - 2 = - \frac{2}{5} x$

Step 1.6.1.1.3

Factor $x$ out of $x (3) + x (- 2 \cdot 5)$ .

$\frac{x (3 - 2 \cdot 5)}{15} - 2 = - \frac{2}{5} x$

Step 1.6.1.2

Multiply $- 2$ by $5$ .

$\frac{x (3 - 10)}{15} - 2 = - \frac{2}{5} x$

Step 1.6.1.3

Subtract $10$ from $3$ .

$\frac{x \cdot - 7}{15} - 2 = - \frac{2}{5} x$

Step 1.6.2

Move $- 7$ to the left of $x$ .

$\frac{- 7 \cdot x}{15} - 2 = - \frac{2}{5} x$

Step 1.6.3

Move the negative in front of the fraction.

$- \frac{7 x}{15} - 2 = - \frac{2}{5} x$

Step 2

Simplify $- \frac{2}{5} x$ .

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Step 2.1

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

$- \frac{7 x}{15} - 2 = - \frac{x \cdot 2}{5}$

Step 2.2

Move $2$ to the left of $x$ .

$- \frac{7 x}{15} - 2 = - \frac{2 x}{5}$

Step 3

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

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Step 3.1

Add $\frac{2 x}{5}$ to both sides of the equation.

$- \frac{7 x}{15} - 2 + \frac{2 x}{5} = 0$

Step 3.2

To write $\frac{2 x}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{7 x}{15} + \frac{2 x}{5} \cdot \frac{3}{3} - 2 = 0$

Step 3.3

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

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

$- \frac{7 x}{15} + \frac{2 x \cdot 3}{5 \cdot 3} - 2 = 0$

Step 3.3.2

Multiply $5$ by $3$ .

$- \frac{7 x}{15} + \frac{2 x \cdot 3}{15} - 2 = 0$

Step 3.4

Combine the numerators over the common denominator.

$\frac{- 7 x + 2 x \cdot 3}{15} - 2 = 0$

Step 3.5

Simplify each term.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Factor $x$ out of $- 7 x + 2 x \cdot 3$ .

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Step 3.5.1.1.1

Factor $x$ out of $- 7 x$ .

$\frac{x \cdot - 7 + 2 x \cdot 3}{15} - 2 = 0$

Step 3.5.1.1.2

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

$\frac{x \cdot - 7 + x (2 \cdot 3)}{15} - 2 = 0$

Step 3.5.1.1.3

Factor $x$ out of $x \cdot - 7 + x (2 \cdot 3)$ .

$\frac{x (- 7 + 2 \cdot 3)}{15} - 2 = 0$

Step 3.5.1.2

Multiply $2$ by $3$ .

$\frac{x (- 7 + 6)}{15} - 2 = 0$

Step 3.5.1.3

Add $- 7$ and $6$ .

$\frac{x \cdot - 1}{15} - 2 = 0$

Step 3.5.2

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x}{15} - 2 = 0$

Step 3.5.3

Move the negative in front of the fraction.

$- \frac{x}{15} - 2 = 0$

Step 4

Add $2$ to both sides of the equation.

$- \frac{x}{15} = 2$

Step 5

Multiply both sides of the equation by $- 15$ .

$- 15 (- \frac{x}{15}) = - 15 \cdot 2$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $- 15 (- \frac{x}{15})$ .

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Step 6.1.1.1

Cancel the common factor of $15$ .

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Step 6.1.1.1.1

Move the leading negative in $- \frac{x}{15}$ into the numerator.

$- 15 \frac{- x}{15} = - 15 \cdot 2$

Step 6.1.1.1.2

Factor $15$ out of $- 15$ .

$15 (- 1) \frac{- x}{15} = - 15 \cdot 2$

Step 6.1.1.1.3

Cancel the common factor.

$15 \cdot - 1 \frac{- x}{15} = - 15 \cdot 2$

Step 6.1.1.1.4

Rewrite the expression.

$- - x = - 15 \cdot 2$

Step 6.1.1.2

Multiply.

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Step 6.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x = - 15 \cdot 2$

Step 6.1.1.2.2

Multiply $x$ by $1$ .

$x = - 15 \cdot 2$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $- 15$ by $2$ .

$x = - 30$