Algebra Examples

$| \frac{2 x - 1}{x + 3} | = \frac{1 - 2 x}{- x - 3}$

Step 1

Multiply both sides by $- x - 3$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $| \frac{2 x - 1}{x + 3} | (- x - 3)$ .

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Step 2.1.1.1

Apply the distributive property.

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

Step 2.1.1.2

Simplify the expression.

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Step 2.1.1.2.1

Rewrite using the commutative property of multiplication.

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

Step 2.1.1.2.2

Move $- 3$ to the left of $| \frac{2 x - 1}{x + 3} |$ .

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

Step 2.1.1.2.3

Reorder factors in $- | \frac{2 x - 1}{x + 3} | x - 3 | \frac{2 x - 1}{x + 3} |$ .

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{1 - 2 x}{- x - 3} (- x - 3)$ .

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Step 2.2.1.1

Multiply $\frac{1 - 2 x}{- x - 3}$ by $- x - 3$ .

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

Step 2.2.1.2

Cancel the common factor of $- x - 3$ .

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Step 2.2.1.2.1

Cancel the common factor.

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

Step 2.2.1.2.2

Divide $1 - 2 x$ by $1$ .

$- x | \frac{2 x - 1}{x + 3} | - 3 | \frac{2 x - 1}{x + 3} | = 1 - 2 x$

Step 2.2.1.3

Reorder $1$ and $- 2 x$ .

$- x | \frac{2 x - 1}{x + 3} | - 3 | \frac{2 x - 1}{x + 3} | = - 2 x + 1$

Step 3

Solve for $x$ .

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Step 3.1

Factor $| \frac{2 x - 1}{x + 3} |$ out of $- x | \frac{2 x - 1}{x + 3} | - 3 | \frac{2 x - 1}{x + 3} |$ .

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Step 3.1.1

Factor $| \frac{2 x - 1}{x + 3} |$ out of $- x | \frac{2 x - 1}{x + 3} |$ .

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

Step 3.1.2

Factor $| \frac{2 x - 1}{x + 3} |$ out of $- 3 | \frac{2 x - 1}{x + 3} |$ .

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

Step 3.1.3

Factor $| \frac{2 x - 1}{x + 3} |$ out of $| \frac{2 x - 1}{x + 3} | (- x) + | \frac{2 x - 1}{x + 3} | \cdot - 3$ .

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

Step 3.2

Divide each term in $| \frac{2 x - 1}{x + 3} | (- x - 3) = - 2 x + 1$ by $- x - 3$ and simplify.

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Step 3.2.1

Divide each term in $| \frac{2 x - 1}{x + 3} | (- x - 3) = - 2 x + 1$ by $- x - 3$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $- x - 3$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide $| \frac{2 x - 1}{x + 3} |$ by $1$ .

$| \frac{2 x - 1}{x + 3} | = \frac{- 2 x}{- x - 3} + \frac{1}{- x - 3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Combine the numerators over the common denominator.

$| \frac{2 x - 1}{x + 3} | = \frac{- 2 x + 1}{- x - 3}$

Step 3.2.3.2

Factor $- 1$ out of $- 2 x$ .

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

Step 3.2.3.3

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 3.2.3.4

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

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

Step 3.2.3.5

Rewrite $- (2 x - 1)$ as $- 1 (2 x - 1)$ .

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

Step 3.2.3.6

Factor $- 1$ out of $- x$ .

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

Step 3.2.3.7

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 3.2.3.8

Factor $- 1$ out of $- (x) - 1 (3)$ .

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

Step 3.2.3.9

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

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

Step 3.2.3.10

Cancel the common factor.

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

Step 3.2.3.11

Rewrite the expression.

$| \frac{2 x - 1}{x + 3} | = \frac{2 x - 1}{x + 3}$

Step 3.3

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

$\frac{2 x - 1}{x + 3} = \pm \frac{2 x - 1}{x + 3}$

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$2 x - 1 = 2 x - 1$

Step 3.4.3

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

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Step 3.4.3.1

Subtract $2 x$ from both sides of the equation.

$2 x - 1 - 2 x = - 1$

Step 3.4.3.2

Combine the opposite terms in $2 x - 1 - 2 x$ .

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Step 3.4.3.2.1

Subtract $2 x$ from $2 x$ .

$0 - 1 = - 1$

Step 3.4.3.2.2

Subtract $1$ from $0$ .

$- 1 = - 1$

Step 3.4.4

Since $- 1 = - 1$ , the equation will always be true.

All real numbers

Step 3.4.5

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

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

Step 3.4.6

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$2 x - 1 = - (2 x - 1)$

Step 3.4.7

Simplify $- (2 x - 1)$ .

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Step 3.4.7.1

Rewrite.

$2 x - 1 = 0 + 0 - (2 x - 1)$

Step 3.4.7.2

Simplify by adding zeros.

$2 x - 1 = - (2 x - 1)$

Step 3.4.7.3

Apply the distributive property.

$2 x - 1 = - (2 x) - - 1$

Step 3.4.7.4

Multiply.

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Step 3.4.7.4.1

Multiply $2$ by $- 1$ .

$2 x - 1 = - 2 x - - 1$

Step 3.4.7.4.2

Multiply $- 1$ by $- 1$ .

$2 x - 1 = - 2 x + 1$

Step 3.4.8

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

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Step 3.4.8.1

Add $2 x$ to both sides of the equation.

$2 x - 1 + 2 x = 1$

Step 3.4.8.2

Add $2 x$ and $2 x$ .

$4 x - 1 = 1$

Step 3.4.9

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

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Step 3.4.9.1

Add $1$ to both sides of the equation.

$4 x = 1 + 1$

Step 3.4.9.2

Add $1$ and $1$ .

$4 x = 2$

Step 3.4.10

Divide each term in $4 x = 2$ by $4$ and simplify.

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Step 3.4.10.1

Divide each term in $4 x = 2$ by $4$ .

$\frac{4 x}{4} = \frac{2}{4}$

Step 3.4.10.2

Simplify the left side.

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Step 3.4.10.2.1

Cancel the common factor of $4$ .

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Step 3.4.10.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{2}{4}$

Step 3.4.10.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{4}$

Step 3.4.10.3

Simplify the right side.

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Step 3.4.10.3.1

Cancel the common factor of $2$ and $4$ .

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Step 3.4.10.3.1.1

Factor $2$ out of $2$ .

$x = \frac{2 (1)}{4}$

Step 3.4.10.3.1.2

Cancel the common factors.

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Step 3.4.10.3.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot 1}{2 \cdot 2}$

Step 3.4.10.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 1}{2 \cdot 2}$

Step 3.4.10.3.1.2.3

Rewrite the expression.

$x = \frac{1}{2}$

Step 3.4.11

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

$x =, \frac{1}{2}$

Step 4

Verify each of the solutions by substituting them into $| \frac{2 x - 1}{x + 3} | = \frac{1 - 2 x}{- x - 3}$ and solving.

$x = \frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}$

Decimal Form:

$x = 0.5$