Basic Math Examples

$4 | 7 y + 2 | - 8 = - 7$

Step 1

Move all terms not containing $| 7 y + 2 |$ to the right side of the equation.

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Step 1.1

Add $8$ to both sides of the equation.

$4 | 7 y + 2 | = - 7 + 8$

Step 1.2

Add $- 7$ and $8$ .

$4 | 7 y + 2 | = 1$

Step 2

Divide each term in $4 | 7 y + 2 | = 1$ by $4$ and simplify.

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Step 2.1

Divide each term in $4 | 7 y + 2 | = 1$ by $4$ .

$\frac{4 | 7 y + 2 |}{4} = \frac{1}{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{4 | 7 y + 2 |}{4} = \frac{1}{4}$

Step 2.2.1.2

Divide $| 7 y + 2 |$ by $1$ .

$| 7 y + 2 | = \frac{1}{4}$

Step 3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$7 y + 2 = \pm \frac{1}{4}$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$7 y + 2 = \frac{1}{4}$

Step 4.2

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

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Step 4.2.1

Subtract $2$ from both sides of the equation.

$7 y = \frac{1}{4} - 2$

Step 4.2.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$7 y = \frac{1}{4} - 2 \cdot \frac{4}{4}$

Step 4.2.3

Combine $- 2$ and $\frac{4}{4}$ .

$7 y = \frac{1}{4} + \frac{- 2 \cdot 4}{4}$

Step 4.2.4

Combine the numerators over the common denominator.

$7 y = \frac{1 - 2 \cdot 4}{4}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 2$ by $4$ .

$7 y = \frac{1 - 8}{4}$

Step 4.2.5.2

Subtract $8$ from $1$ .

$7 y = \frac{- 7}{4}$

Step 4.2.6

Move the negative in front of the fraction.

$7 y = - \frac{7}{4}$

Step 4.3

Divide each term in $7 y = - \frac{7}{4}$ by $7$ and simplify.

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Step 4.3.1

Divide each term in $7 y = - \frac{7}{4}$ by $7$ .

$\frac{7 y}{7} = \frac{- \frac{7}{4}}{7}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $7$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{- \frac{7}{4}}{7}$

Step 4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{7}{4}}{7}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{7}{4} \cdot \frac{1}{7}$

Step 4.3.3.2

Cancel the common factor of $7$ .

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Step 4.3.3.2.1

Move the leading negative in $- \frac{7}{4}$ into the numerator.

$y = \frac{- 7}{4} \cdot \frac{1}{7}$

Step 4.3.3.2.2

Factor $7$ out of $- 7$ .

$y = \frac{7 (- 1)}{4} \cdot \frac{1}{7}$

Step 4.3.3.2.3

Cancel the common factor.

$y = \frac{7 \cdot - 1}{4} \cdot \frac{1}{7}$

Step 4.3.3.2.4

Rewrite the expression.

$y = \frac{- 1}{4}$

Step 4.3.3.3

Move the negative in front of the fraction.

$y = - \frac{1}{4}$

Step 4.4

Next, use the negative value of the $\pm$ to find the second solution.

$7 y + 2 = - \frac{1}{4}$

Step 4.5

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

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Step 4.5.1

Subtract $2$ from both sides of the equation.

$7 y = - \frac{1}{4} - 2$

Step 4.5.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$7 y = - \frac{1}{4} - 2 \cdot \frac{4}{4}$

Step 4.5.3

Combine $- 2$ and $\frac{4}{4}$ .

$7 y = - \frac{1}{4} + \frac{- 2 \cdot 4}{4}$

Step 4.5.4

Combine the numerators over the common denominator.

$7 y = \frac{- 1 - 2 \cdot 4}{4}$

Step 4.5.5

Simplify the numerator.

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Step 4.5.5.1

Multiply $- 2$ by $4$ .

$7 y = \frac{- 1 - 8}{4}$

Step 4.5.5.2

Subtract $8$ from $- 1$ .

$7 y = \frac{- 9}{4}$

Step 4.5.6

Move the negative in front of the fraction.

$7 y = - \frac{9}{4}$

Step 4.6

Divide each term in $7 y = - \frac{9}{4}$ by $7$ and simplify.

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Step 4.6.1

Divide each term in $7 y = - \frac{9}{4}$ by $7$ .

$\frac{7 y}{7} = \frac{- \frac{9}{4}}{7}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $7$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{- \frac{9}{4}}{7}$

Step 4.6.2.1.2

Divide $y$ by $1$ .

$y = \frac{- \frac{9}{4}}{7}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{9}{4} \cdot \frac{1}{7}$

Step 4.6.3.2

Multiply $- \frac{9}{4} \cdot \frac{1}{7}$ .

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Step 4.6.3.2.1

Multiply $\frac{1}{7}$ by $\frac{9}{4}$ .

$y = - \frac{9}{7 \cdot 4}$

Step 4.6.3.2.2

Multiply $7$ by $4$ .

$y = - \frac{9}{28}$

Step 4.7

The complete solution is the result of both the positive and negative portions of the solution.

$y = - \frac{1}{4}, - \frac{9}{28}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{1}{4}, - \frac{9}{28}$

Decimal Form:

$y = - 0.25, - 0.32142857 \dots$