Algebra Examples

$\frac{5}{8} (4 x - 12) + \frac{1}{2} x - 3 \frac{1}{2} = 3 (x + 3)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\frac{5}{8} \cdot (4 x - 12) + \frac{1}{2} x - 3 \frac{1}{2}$ .

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Step 1.1.1

Convert $3 \frac{1}{2}$ to an improper fraction.

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Step 1.1.1.1

A mixed number is an addition of its whole and fractional parts.

$\frac{5}{8} \cdot (4 x - 12) + \frac{1}{2} x - (3 + \frac{1}{2}) = 3 (x + 3)$

Step 1.1.1.2

Add $3$ and $\frac{1}{2}$ .

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Step 1.1.1.2.1

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

$\frac{5}{8} \cdot (4 x - 12) + \frac{1}{2} x - (3 \cdot \frac{2}{2} + \frac{1}{2}) = 3 (x + 3)$

Step 1.1.1.2.2

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

$\frac{5}{8} \cdot (4 x - 12) + \frac{1}{2} x - (\frac{3 \cdot 2}{2} + \frac{1}{2}) = 3 (x + 3)$

Step 1.1.1.2.3

Combine the numerators over the common denominator.

$\frac{5}{8} \cdot (4 x - 12) + \frac{1}{2} x - \frac{3 \cdot 2 + 1}{2} = 3 (x + 3)$

Step 1.1.1.2.4

Simplify the numerator.

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Step 1.1.1.2.4.1

Multiply $3$ by $2$ .

$\frac{5}{8} \cdot (4 x - 12) + \frac{1}{2} x - \frac{6 + 1}{2} = 3 (x + 3)$

Step 1.1.1.2.4.2

Add $6$ and $1$ .

$\frac{5}{8} \cdot (4 x - 12) + \frac{1}{2} x - \frac{7}{2} = 3 (x + 3)$

Step 1.1.2

Simplify each term.

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Step 1.1.2.1

Apply the distributive property.

$\frac{5}{8} (4 x) + \frac{5}{8} \cdot - 12 + \frac{1}{2} x - \frac{7}{2} = 3 (x + 3)$

Step 1.1.2.2

Cancel the common factor of $4$ .

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Step 1.1.2.2.1

Factor $4$ out of $8$ .

$\frac{5}{4 (2)} (4 x) + \frac{5}{8} \cdot - 12 + \frac{1}{2} x - \frac{7}{2} = 3 (x + 3)$

Step 1.1.2.2.2

Factor $4$ out of $4 x$ .

$\frac{5}{4 (2)} (4 (x)) + \frac{5}{8} \cdot - 12 + \frac{1}{2} x - \frac{7}{2} = 3 (x + 3)$

Step 1.1.2.2.3

Cancel the common factor.

$\frac{5}{4 \cdot 2} (4 x) + \frac{5}{8} \cdot - 12 + \frac{1}{2} x - \frac{7}{2} = 3 (x + 3)$

Step 1.1.2.2.4

Rewrite the expression.

$\frac{5}{2} x + \frac{5}{8} \cdot - 12 + \frac{1}{2} x - \frac{7}{2} = 3 (x + 3)$

Step 1.1.2.3

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

$\frac{5 x}{2} + \frac{5}{8} \cdot - 12 + \frac{1}{2} x - \frac{7}{2} = 3 (x + 3)$

Step 1.1.2.4

Cancel the common factor of $4$ .

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Step 1.1.2.4.1

Factor $4$ out of $8$ .

$\frac{5 x}{2} + \frac{5}{4 (2)} \cdot - 12 + \frac{1}{2} x - \frac{7}{2} = 3 (x + 3)$

Step 1.1.2.4.2

Factor $4$ out of $- 12$ .

$\frac{5 x}{2} + \frac{5}{4 \cdot 2} \cdot (4 \cdot - 3) + \frac{1}{2} x - \frac{7}{2} = 3 (x + 3)$

Step 1.1.2.4.3

Cancel the common factor.

$\frac{5 x}{2} + \frac{5}{4 \cdot 2} \cdot (4 \cdot - 3) + \frac{1}{2} x - \frac{7}{2} = 3 (x + 3)$

Step 1.1.2.4.4

Rewrite the expression.

$\frac{5 x}{2} + \frac{5}{2} \cdot - 3 + \frac{1}{2} x - \frac{7}{2} = 3 (x + 3)$

Step 1.1.2.5

Combine $\frac{5}{2}$ and $- 3$ .

$\frac{5 x}{2} + \frac{5 \cdot - 3}{2} + \frac{1}{2} x - \frac{7}{2} = 3 (x + 3)$

Step 1.1.2.6

Multiply $5$ by $- 3$ .

$\frac{5 x}{2} + \frac{- 15}{2} + \frac{1}{2} x - \frac{7}{2} = 3 (x + 3)$

Step 1.1.2.7

Move the negative in front of the fraction.

$\frac{5 x}{2} - \frac{15}{2} + \frac{1}{2} x - \frac{7}{2} = 3 (x + 3)$

Step 1.1.2.8

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

$\frac{5 x}{2} - \frac{15}{2} + \frac{x}{2} - \frac{7}{2} = 3 (x + 3)$

Step 1.1.3

Simplify terms.

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Step 1.1.3.1

Combine the numerators over the common denominator.

$\frac{5 x - 15 + x - 7}{2} = 3 (x + 3)$

Step 1.1.3.2

Add $5 x$ and $x$ .

$\frac{6 x - 15 - 7}{2} = 3 (x + 3)$

Step 1.1.3.3

Subtract $7$ from $- 15$ .

$\frac{6 x - 22}{2} = 3 (x + 3)$

Step 1.1.3.4

Cancel the common factor of $6 x - 22$ and $2$ .

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Step 1.1.3.4.1

Factor $2$ out of $6 x$ .

$\frac{2 (3 x) - 22}{2} = 3 (x + 3)$

Step 1.1.3.4.2

Factor $2$ out of $- 22$ .

$\frac{2 (3 x) + 2 \cdot - 11}{2} = 3 (x + 3)$

Step 1.1.3.4.3

Factor $2$ out of $2 (3 x) + 2 (- 11)$ .

$\frac{2 (3 x - 11)}{2} = 3 (x + 3)$

Step 1.1.3.4.4

Cancel the common factors.

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Step 1.1.3.4.4.1

Factor $2$ out of $2$ .

$\frac{2 (3 x - 11)}{2 (1)} = 3 (x + 3)$

Step 1.1.3.4.4.2

Cancel the common factor.

$\frac{2 (3 x - 11)}{2 \cdot 1} = 3 (x + 3)$

Step 1.1.3.4.4.3

Rewrite the expression.

$\frac{3 x - 11}{1} = 3 (x + 3)$

Step 1.1.3.4.4.4

Divide $3 x - 11$ by $1$ .

$3 x - 11 = 3 (x + 3)$

Step 2

Simplify the right side.

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Step 2.1

Simplify $3 (x + 3)$ .

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Step 2.1.1

Apply the distributive property.

$3 x - 11 = 3 x + 3 \cdot 3$

Step 2.1.2

Multiply $3$ by $3$ .

$3 x - 11 = 3 x + 9$

Step 3

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

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Step 3.1

Subtract $3 x$ from both sides of the equation.

$3 x - 11 - 3 x = 9$

Step 3.2

Combine the opposite terms in $3 x - 11 - 3 x$ .

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Step 3.2.1

Subtract $3 x$ from $3 x$ .

$0 - 11 = 9$

Step 3.2.2

Subtract $11$ from $0$ .

$- 11 = 9$

Step 4

Since $- 11 \neq 9$ , there are no solutions.

No solution