Pre-Algebra Examples

$- \frac{1}{2} + \frac{3}{4} \cdot (x - 1) = \frac{5}{2}$

Step 1

Simplify $- \frac{1}{2} + \frac{3}{4} \cdot (x - 1)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$- \frac{1}{2} + \frac{3}{4} x + \frac{3}{4} \cdot - 1 = \frac{5}{2}$

Step 1.1.2

Combine $\frac{3}{4}$ and $x$ .

$- \frac{1}{2} + \frac{3 x}{4} + \frac{3}{4} \cdot - 1 = \frac{5}{2}$

Step 1.1.3

Multiply $\frac{3}{4} \cdot - 1$ .

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Step 1.1.3.1

Combine $\frac{3}{4}$ and $- 1$ .

$- \frac{1}{2} + \frac{3 x}{4} + \frac{3 \cdot - 1}{4} = \frac{5}{2}$

Step 1.1.3.2

Multiply $3$ by $- 1$ .

$- \frac{1}{2} + \frac{3 x}{4} + \frac{- 3}{4} = \frac{5}{2}$

Step 1.1.4

Move the negative in front of the fraction.

$- \frac{1}{2} + \frac{3 x}{4} - \frac{3}{4} = \frac{5}{2}$

Step 1.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 x}{4} - \frac{1}{2} \cdot \frac{2}{2} - \frac{3}{4} = \frac{5}{2}$

Step 1.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

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

$\frac{3 x}{4} - \frac{2}{2 \cdot 2} - \frac{3}{4} = \frac{5}{2}$

Step 1.3.2

Multiply $2$ by $2$ .

$\frac{3 x}{4} - \frac{2}{4} - \frac{3}{4} = \frac{5}{2}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{3 x}{4} + \frac{- 2 - 3}{4} = \frac{5}{2}$

Step 1.5

Subtract $3$ from $- 2$ .

$\frac{3 x}{4} + \frac{- 5}{4} = \frac{5}{2}$

Step 1.6

Move the negative in front of the fraction.

$\frac{3 x}{4} - \frac{5}{4} = \frac{5}{2}$

Step 2

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

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Step 2.1

Add $\frac{5}{4}$ to both sides of the equation.

$\frac{3 x}{4} = \frac{5}{2} + \frac{5}{4}$

Step 2.2

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

$\frac{3 x}{4} = \frac{5}{2} \cdot \frac{2}{2} + \frac{5}{4}$

Step 2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

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

$\frac{3 x}{4} = \frac{5 \cdot 2}{2 \cdot 2} + \frac{5}{4}$

Step 2.3.2

Multiply $2$ by $2$ .

$\frac{3 x}{4} = \frac{5 \cdot 2}{4} + \frac{5}{4}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3 x}{4} = \frac{5 \cdot 2 + 5}{4}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $5$ by $2$ .

$\frac{3 x}{4} = \frac{10 + 5}{4}$

Step 2.5.2

Add $10$ and $5$ .

$\frac{3 x}{4} = \frac{15}{4}$

Step 3

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

$3 x = 15$

Step 4

Divide each term in $3 x = 15$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 x = 15$ by $3$ .

$\frac{3 x}{3} = \frac{15}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{15}{3}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{15}{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $15$ by $3$ .

$x = 5$