Algebra Examples

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

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

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:

$x = 0.5$