Finite Math Examples

$| \frac{3 x - 5}{2} | = 7$

Step 1

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

$\frac{3 x - 5}{2} = \pm 7$

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.

$\frac{3 x - 5}{2} = 7$

Step 2.2

Multiply both sides by $2$ .

$\frac{3 x - 5}{2} \cdot 2 = 7 \cdot 2$

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 $2$ .

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Step 2.3.1.1.1

Cancel the common factor.

$\frac{3 x - 5}{2} \cdot 2 = 7 \cdot 2$

Step 2.3.1.1.2

Rewrite the expression.

$3 x - 5 = 7 \cdot 2$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $7$ by $2$ .

$3 x - 5 = 14$

Step 2.4

Solve for $x$ .

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Step 2.4.1

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

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Step 2.4.1.1

Add $5$ to both sides of the equation.

$3 x = 14 + 5$

Step 2.4.1.2

Add $14$ and $5$ .

$3 x = 19$

Step 2.4.2

Divide each term in $3 x = 19$ by $3$ and simplify.

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Step 2.4.2.1

Divide each term in $3 x = 19$ by $3$ .

$\frac{3 x}{3} = \frac{19}{3}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{19}{3}$

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{19}{3}$

Step 2.5

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

$\frac{3 x - 5}{2} = - 7$

Step 2.6

Multiply both sides by $2$ .

$\frac{3 x - 5}{2} \cdot 2 = - 7 \cdot 2$

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 $2$ .

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Step 2.7.1.1.1

Cancel the common factor.

$\frac{3 x - 5}{2} \cdot 2 = - 7 \cdot 2$

Step 2.7.1.1.2

Rewrite the expression.

$3 x - 5 = - 7 \cdot 2$

Step 2.7.2

Simplify the right side.

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Step 2.7.2.1

Multiply $- 7$ by $2$ .

$3 x - 5 = - 14$

Step 2.8

Solve for $x$ .

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Step 2.8.1

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

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Step 2.8.1.1

Add $5$ to both sides of the equation.

$3 x = - 14 + 5$

Step 2.8.1.2

Add $- 14$ and $5$ .

$3 x = - 9$

Step 2.8.2

Divide each term in $3 x = - 9$ by $3$ and simplify.

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Step 2.8.2.1

Divide each term in $3 x = - 9$ by $3$ .

$\frac{3 x}{3} = \frac{- 9}{3}$

Step 2.8.2.2

Simplify the left side.

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Step 2.8.2.2.1

Cancel the common factor of $3$ .

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Step 2.8.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 9}{3}$

Step 2.8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{3}$

Step 2.8.2.3

Simplify the right side.

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Step 2.8.2.3.1

Divide $- 9$ by $3$ .

$x = - 3$

Step 2.9

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

$x = \frac{19}{3}, - 3$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{19}{3}, - 3$

Decimal Form:

$x = 6.\overline{3}, - 3$

Mixed Number Form:

$x = 6 \frac{1}{3}, - 3$