Precalculus Examples

$\frac{5 x}{3} - x = \frac{x}{6} - \frac{3}{2}$

Step 1

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

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Step 1.1

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

$\frac{5 x}{3} - x \cdot \frac{3}{3} = \frac{x}{6} - \frac{3}{2}$

Step 1.2

Simplify terms.

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Step 1.2.1

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

$\frac{5 x}{3} + \frac{- x \cdot 3}{3} = \frac{x}{6} - \frac{3}{2}$

Step 1.2.2

Combine the numerators over the common denominator.

$\frac{5 x - x \cdot 3}{3} = \frac{x}{6} - \frac{3}{2}$

Step 1.3

Simplify the numerator.

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Step 1.3.1

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

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Step 1.3.1.1

Factor $x$ out of $5 x$ .

$\frac{x \cdot 5 - x \cdot 3}{3} = \frac{x}{6} - \frac{3}{2}$

Step 1.3.1.2

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

$\frac{x \cdot 5 + x (- 1 \cdot 3)}{3} = \frac{x}{6} - \frac{3}{2}$

Step 1.3.1.3

Factor $x$ out of $x \cdot 5 + x (- 1 \cdot 3)$ .

$\frac{x (5 - 1 \cdot 3)}{3} = \frac{x}{6} - \frac{3}{2}$

Step 1.3.2

Multiply $- 1$ by $3$ .

$\frac{x (5 - 3)}{3} = \frac{x}{6} - \frac{3}{2}$

Step 1.3.3

Subtract $3$ from $5$ .

$\frac{x \cdot 2}{3} = \frac{x}{6} - \frac{3}{2}$

Step 1.4

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

$\frac{2 x}{3} = \frac{x}{6} - \frac{3}{2}$

Step 2

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

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Step 2.1

Subtract $\frac{x}{6}$ from both sides of the equation.

$\frac{2 x}{3} - \frac{x}{6} = - \frac{3}{2}$

Step 2.2

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

$\frac{2 x}{3} \cdot \frac{2}{2} - \frac{x}{6} = - \frac{3}{2}$

Step 2.3

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

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Step 2.3.1

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

$\frac{2 x \cdot 2}{3 \cdot 2} - \frac{x}{6} = - \frac{3}{2}$

Step 2.3.2

Multiply $3$ by $2$ .

$\frac{2 x \cdot 2}{6} - \frac{x}{6} = - \frac{3}{2}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{2 x \cdot 2 - x}{6} = - \frac{3}{2}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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Step 2.5.1.1

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

$\frac{x (2 \cdot 2) - x}{6} = - \frac{3}{2}$

Step 2.5.1.2

Factor $x$ out of $- x$ .

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

Step 2.5.1.3

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

$\frac{x (2 \cdot 2 - 1)}{6} = - \frac{3}{2}$

Step 2.5.2

Multiply $2$ by $2$ .

$\frac{x (4 - 1)}{6} = - \frac{3}{2}$

Step 2.5.3

Subtract $1$ from $4$ .

$\frac{x \cdot 3}{6} = - \frac{3}{2}$

Step 2.6

Cancel the common factor of $3$ and $6$ .

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Step 2.6.1

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

$\frac{3 \cdot x}{6} = - \frac{3}{2}$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $3$ out of $6$ .

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

Step 2.6.2.2

Cancel the common factor.

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

Step 2.6.2.3

Rewrite the expression.

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

Step 3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = - 3$