Precalculus Examples

$- \frac{7}{2} \cdot (x + 6) + 7 x = - 42$

Step 1

Simplify $- \frac{7}{2} \cdot (x + 6) + 7 x$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$- \frac{7}{2} x - \frac{7}{2} \cdot 6 + 7 x = - 42$

Step 1.1.2

Combine $x$ and $\frac{7}{2}$ .

$- \frac{x \cdot 7}{2} - \frac{7}{2} \cdot 6 + 7 x = - 42$

Step 1.1.3

Cancel the common factor of $2$ .

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Step 1.1.3.1

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$- \frac{x \cdot 7}{2} + \frac{- 7}{2} \cdot 6 + 7 x = - 42$

Step 1.1.3.2

Factor $2$ out of $6$ .

$- \frac{x \cdot 7}{2} + \frac{- 7}{2} \cdot (2 (3)) + 7 x = - 42$

Step 1.1.3.3

Cancel the common factor.

$- \frac{x \cdot 7}{2} + \frac{- 7}{2} \cdot (2 \cdot 3) + 7 x = - 42$

Step 1.1.3.4

Rewrite the expression.

$- \frac{x \cdot 7}{2} - 7 \cdot 3 + 7 x = - 42$

Step 1.1.4

Multiply $- 7$ by $3$ .

$- \frac{x \cdot 7}{2} - 21 + 7 x = - 42$

Step 1.1.5

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

$- \frac{7 x}{2} - 21 + 7 x = - 42$

Step 1.2

To write $7 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{7 x}{2} + 7 x \cdot \frac{2}{2} - 21 = - 42$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $7 x$ and $\frac{2}{2}$ .

$- \frac{7 x}{2} + \frac{7 x \cdot 2}{2} - 21 = - 42$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{- 7 x + 7 x \cdot 2}{2} - 21 = - 42$

Step 1.4

Simplify the numerator.

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Step 1.4.1

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

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Step 1.4.1.1

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

$\frac{7 x (- 1) + 7 x \cdot 2}{2} - 21 = - 42$

Step 1.4.1.2

Factor $7 x$ out of $7 x \cdot 2$ .

$\frac{7 x (- 1) + 7 x (2)}{2} - 21 = - 42$

Step 1.4.1.3

Factor $7 x$ out of $7 x (- 1) + 7 x (2)$ .

$\frac{7 x (- 1 + 2)}{2} - 21 = - 42$

Step 1.4.2

Add $- 1$ and $2$ .

$\frac{7 x \cdot 1}{2} - 21 = - 42$

Step 1.4.3

Multiply $7$ by $1$ .

$\frac{7 x}{2} - 21 = - 42$

Step 2

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

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Step 2.1

Add $21$ to both sides of the equation.

$\frac{7 x}{2} = - 42 + 21$

Step 2.2

Add $- 42$ and $21$ .

$\frac{7 x}{2} = - 21$

Step 3

Multiply both sides of the equation by $\frac{2}{7}$ .

$\frac{2}{7} \cdot \frac{7 x}{2} = \frac{2}{7} \cdot - 21$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{2}{7} \cdot \frac{7 x}{2}$ .

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Step 4.1.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{2}{7} \cdot \frac{7 x}{2} = \frac{2}{7} \cdot - 21$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{7} (7 x) = \frac{2}{7} \cdot - 21$

Step 4.1.1.2

Cancel the common factor of $7$ .

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Step 4.1.1.2.1

Factor $7$ out of $7 x$ .

$\frac{1}{7} (7 (x)) = \frac{2}{7} \cdot - 21$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{7} (7 x) = \frac{2}{7} \cdot - 21$

Step 4.1.1.2.3

Rewrite the expression.

$x = \frac{2}{7} \cdot - 21$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{2}{7} \cdot - 21$ .

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Step 4.2.1.1

Cancel the common factor of $7$ .

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Step 4.2.1.1.1

Factor $7$ out of $- 21$ .

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

Step 4.2.1.1.2

Cancel the common factor.

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

Step 4.2.1.1.3

Rewrite the expression.

$x = 2 \cdot - 3$

Step 4.2.1.2

Multiply $2$ by $- 3$ .

$x = - 6$