Finite Math Examples

$| \frac{7 x - 10}{8} | = 7 x + 1$

Step 1

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

$\frac{7 x - 10}{8} = \pm (7 x + 1)$

Step 2

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

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Step 2.1

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

$\frac{7 x - 10}{8} = 7 x + 1$

Step 2.2

Multiply both sides by $8$ .

$\frac{7 x - 10}{8} \cdot 8 = (7 x + 1) \cdot 8$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Cancel the common factor of $8$ .

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Step 2.3.1.1.1

Cancel the common factor.

$\frac{7 x - 10}{8} \cdot 8 = (7 x + 1) \cdot 8$

Step 2.3.1.1.2

Rewrite the expression.

$7 x - 10 = (7 x + 1) \cdot 8$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $(7 x + 1) \cdot 8$ .

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Step 2.3.2.1.1

Apply the distributive property.

$7 x - 10 = 7 x \cdot 8 + 1 \cdot 8$

Step 2.3.2.1.2

Multiply.

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Step 2.3.2.1.2.1

Multiply $8$ by $7$ .

$7 x - 10 = 56 x + 1 \cdot 8$

Step 2.3.2.1.2.2

Multiply $8$ by $1$ .

$7 x - 10 = 56 x + 8$

Step 2.4

Solve for $x$ .

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Step 2.4.1

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

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Step 2.4.1.1

Subtract $56 x$ from both sides of the equation.

$7 x - 10 - 56 x = 8$

Step 2.4.1.2

Subtract $56 x$ from $7 x$ .

$- 49 x - 10 = 8$

Step 2.4.2

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

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Step 2.4.2.1

Add $10$ to both sides of the equation.

$- 49 x = 8 + 10$

Step 2.4.2.2

Add $8$ and $10$ .

$- 49 x = 18$

Step 2.4.3

Divide each term in $- 49 x = 18$ by $- 49$ and simplify.

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Step 2.4.3.1

Divide each term in $- 49 x = 18$ by $- 49$ .

$\frac{- 49 x}{- 49} = \frac{18}{- 49}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $- 49$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

$\frac{- 49 x}{- 49} = \frac{18}{- 49}$

Step 2.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{18}{- 49}$

Step 2.4.3.3

Simplify the right side.

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Step 2.4.3.3.1

Move the negative in front of the fraction.

$x = - \frac{18}{49}$

Step 2.5

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

$\frac{7 x - 10}{8} = - (7 x + 1)$

Step 2.6

Multiply both sides by $8$ .

$\frac{7 x - 10}{8} \cdot 8 = - (7 x + 1) \cdot 8$

Step 2.7

Simplify.

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Step 2.7.1

Simplify the left side.

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Step 2.7.1.1

Cancel the common factor of $8$ .

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Step 2.7.1.1.1

Cancel the common factor.

$\frac{7 x - 10}{8} \cdot 8 = - (7 x + 1) \cdot 8$

Step 2.7.1.1.2

Rewrite the expression.

$7 x - 10 = - (7 x + 1) \cdot 8$

Step 2.7.2

Simplify the right side.

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Step 2.7.2.1

Simplify $- (7 x + 1) \cdot 8$ .

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Step 2.7.2.1.1

Apply the distributive property.

$7 x - 10 = (- (7 x) - 1 \cdot 1) \cdot 8$

Step 2.7.2.1.2

Multiply.

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Step 2.7.2.1.2.1

Multiply $7$ by $- 1$ .

$7 x - 10 = (- 7 x - 1 \cdot 1) \cdot 8$

Step 2.7.2.1.2.2

Multiply $- 1$ by $1$ .

$7 x - 10 = (- 7 x - 1) \cdot 8$

Step 2.7.2.1.3

Apply the distributive property.

$7 x - 10 = - 7 x \cdot 8 - 1 \cdot 8$

Step 2.7.2.1.4

Multiply.

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Step 2.7.2.1.4.1

Multiply $8$ by $- 7$ .

$7 x - 10 = - 56 x - 1 \cdot 8$

Step 2.7.2.1.4.2

Multiply $- 1$ by $8$ .

$7 x - 10 = - 56 x - 8$

Step 2.8

Solve for $x$ .

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Step 2.8.1

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

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Step 2.8.1.1

Add $56 x$ to both sides of the equation.

$7 x - 10 + 56 x = - 8$

Step 2.8.1.2

Add $7 x$ and $56 x$ .

$63 x - 10 = - 8$

Step 2.8.2

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

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Step 2.8.2.1

Add $10$ to both sides of the equation.

$63 x = - 8 + 10$

Step 2.8.2.2

Add $- 8$ and $10$ .

$63 x = 2$

Step 2.8.3

Divide each term in $63 x = 2$ by $63$ and simplify.

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Step 2.8.3.1

Divide each term in $63 x = 2$ by $63$ .

$\frac{63 x}{63} = \frac{2}{63}$

Step 2.8.3.2

Simplify the left side.

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Step 2.8.3.2.1

Cancel the common factor of $63$ .

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Step 2.8.3.2.1.1

Cancel the common factor.

$\frac{63 x}{63} = \frac{2}{63}$

Step 2.8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{63}$

Step 2.9

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

$x = - \frac{18}{49}, \frac{2}{63}$

Step 3

Exclude the solutions that do not make $| \frac{7 x - 10}{8} | = 7 x + 1$ true.

$x = \frac{2}{63}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{63}$

Decimal Form:

$x = 0.\overline{031746}$