Algebra Examples

$| 4 x | = | 15 + x | | 5 | | 2 y - 1 |$

Step 1

Rewrite the equation as $| 15 + x | | 5 | | 2 y - 1 | = | 4 x |$ .

$| 15 + x | | 5 | | 2 y - 1 | = | 4 x |$

Step 2

Divide each term in $| 15 + x | | 5 | | 2 y - 1 | = | 4 x |$ by $| 5 | | 2 y - 1 |$ and simplify.

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Step 2.1

Divide each term in $| 15 + x | | 5 | | 2 y - 1 | = | 4 x |$ by $| 5 | | 2 y - 1 |$ .

$\frac{| 15 + x | | 5 | | 2 y - 1 |}{| 5 | | 2 y - 1 |} = \frac{| 4 x |}{| 5 | | 2 y - 1 |}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $| 5 |$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{| 15 + x | | 5 | | 2 y - 1 |}{| 5 | | 2 y - 1 |} = \frac{| 4 x |}{| 5 | | 2 y - 1 |}$

Step 2.2.1.2

Rewrite the expression.

$\frac{| 15 + x | | 2 y - 1 |}{| 2 y - 1 |} = \frac{| 4 x |}{| 5 | | 2 y - 1 |}$

Step 2.2.2

Cancel the common factor of $| 2 y - 1 |$ .

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Step 2.2.2.1

Cancel the common factor.

$\frac{| 15 + x | | 2 y - 1 |}{| 2 y - 1 |} = \frac{| 4 x |}{| 5 | | 2 y - 1 |}$

Step 2.2.2.2

Divide $| 15 + x |$ by $1$ .

$| 15 + x | = \frac{| 4 x |}{| 5 | | 2 y - 1 |}$

Step 2.3

Simplify the right side.

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Step 2.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$| 15 + x | = \frac{| 4 x |}{5 | 2 y - 1 |}$

Step 3

Rewrite the absolute value equation as four equations without absolute value bars.

$15 + x = \frac{4 x}{5 | 2 y - 1 |}$

$15 + x = - \frac{4 x}{5 | 2 y - 1 |}$

$- (15 + x) = \frac{4 x}{5 | 2 y - 1 |}$

$- (15 + x) = - \frac{4 x}{5 | 2 y - 1 |}$

Step 4

After simplifying, there are only two unique equations to be solved.

$15 + x = \frac{4 x}{5 | 2 y - 1 |}$

$15 + x = - \frac{4 x}{5 | 2 y - 1 |}$

Step 5

Solve $15 + x = \frac{4 x}{5 | 2 y - 1 |}$ for $x$ .

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Step 5.1

Multiply both sides by $5 | 2 y - 1 |$ .

$(15 + x) (5 | 2 y - 1 |) = \frac{4 x}{5 | 2 y - 1 |} (5 | 2 y - 1 |)$

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $(15 + x) (5 | 2 y - 1 |)$ .

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Step 5.2.1.1.1

Apply the distributive property.

$15 (5 | 2 y - 1 |) + x (5 | 2 y - 1 |) = \frac{4 x}{5 | 2 y - 1 |} (5 | 2 y - 1 |)$

Step 5.2.1.1.2

Simplify the expression.

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Step 5.2.1.1.2.1

Multiply $5$ by $15$ .

$75 | 2 y - 1 | + x (5 | 2 y - 1 |) = \frac{4 x}{5 | 2 y - 1 |} (5 | 2 y - 1 |)$

Step 5.2.1.1.2.2

Rewrite using the commutative property of multiplication.

$75 | 2 y - 1 | + 5 x | 2 y - 1 | = \frac{4 x}{5 | 2 y - 1 |} (5 | 2 y - 1 |)$

Step 5.2.1.1.2.3

Reorder $75 | 2 y - 1 |$ and $5 x | 2 y - 1 |$ .

$5 x | 2 y - 1 | + 75 | 2 y - 1 | = \frac{4 x}{5 | 2 y - 1 |} (5 | 2 y - 1 |)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $\frac{4 x}{5 | 2 y - 1 |} (5 | 2 y - 1 |)$ .

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Step 5.2.2.1.1

Rewrite using the commutative property of multiplication.

$5 x | 2 y - 1 | + 75 | 2 y - 1 | = 5 \frac{4 x}{5 | 2 y - 1 |} | 2 y - 1 |$

Step 5.2.2.1.2

Cancel the common factor of $5$ .

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Step 5.2.2.1.2.1

Factor $5$ out of $5 | 2 y - 1 |$ .

$5 x | 2 y - 1 | + 75 | 2 y - 1 | = 5 \frac{4 x}{5 (| 2 y - 1 |)} | 2 y - 1 |$

Step 5.2.2.1.2.2

Cancel the common factor.

$5 x | 2 y - 1 | + 75 | 2 y - 1 | = 5 \frac{4 x}{5 | 2 y - 1 |} | 2 y - 1 |$

Step 5.2.2.1.2.3

Rewrite the expression.

$5 x | 2 y - 1 | + 75 | 2 y - 1 | = \frac{4 x}{| 2 y - 1 |} | 2 y - 1 |$

Step 5.2.2.1.3

Cancel the common factor of $| 2 y - 1 |$ .

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Step 5.2.2.1.3.1

Cancel the common factor.

$5 x | 2 y - 1 | + 75 | 2 y - 1 | = \frac{4 x}{| 2 y - 1 |} | 2 y - 1 |$

Step 5.2.2.1.3.2

Rewrite the expression.

$5 x | 2 y - 1 | + 75 | 2 y - 1 | = 4 x$

Step 5.3

Solve for $x$ .

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Step 5.3.1

Subtract $4 x$ from both sides of the equation.

$5 x | 2 y - 1 | + 75 | 2 y - 1 | - 4 x = 0$

Step 5.3.2

Subtract $75 | 2 y - 1 |$ from both sides of the equation.

$5 x | 2 y - 1 | - 4 x = - 75 | 2 y - 1 |$

Step 5.3.3

Factor $x$ out of $5 x | 2 y - 1 | - 4 x$ .

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Step 5.3.3.1

Factor $x$ out of $5 x | 2 y - 1 |$ .

$x (5 | 2 y - 1 |) - 4 x = - 75 | 2 y - 1 |$

Step 5.3.3.2

Factor $x$ out of $- 4 x$ .

$x (5 | 2 y - 1 |) + x \cdot - 4 = - 75 | 2 y - 1 |$

Step 5.3.3.3

Factor $x$ out of $x (5 | 2 y - 1 |) + x \cdot - 4$ .

$x (5 | 2 y - 1 | - 4) = - 75 | 2 y - 1 |$

Step 5.3.4

Divide each term in $x (5 | 2 y - 1 | - 4) = - 75 | 2 y - 1 |$ by $5 | 2 y - 1 | - 4$ and simplify.

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Step 5.3.4.1

Divide each term in $x (5 | 2 y - 1 | - 4) = - 75 | 2 y - 1 |$ by $5 | 2 y - 1 | - 4$ .

$\frac{x (5 | 2 y - 1 | - 4)}{5 | 2 y - 1 | - 4} = \frac{- 75 | 2 y - 1 |}{5 | 2 y - 1 | - 4}$

Step 5.3.4.2

Simplify the left side.

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Step 5.3.4.2.1

Cancel the common factor of $5 | 2 y - 1 | - 4$ .

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Step 5.3.4.2.1.1

Cancel the common factor.

$\frac{x (5 | 2 y - 1 | - 4)}{5 | 2 y - 1 | - 4} = \frac{- 75 | 2 y - 1 |}{5 | 2 y - 1 | - 4}$

Step 5.3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 75 | 2 y - 1 |}{5 | 2 y - 1 | - 4}$

Step 5.3.4.3

Simplify the right side.

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Step 5.3.4.3.1

Move the negative in front of the fraction.

$x = - \frac{75 | 2 y - 1 |}{5 | 2 y - 1 | - 4}$

Step 6

Solve $15 + x = - \frac{4 x}{5 | 2 y - 1 |}$ for $x$ .

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Step 6.1

Multiply both sides by $5 | 2 y - 1 |$ .

$(15 + x) (5 | 2 y - 1 |) = - \frac{4 x}{5 | 2 y - 1 |} (5 | 2 y - 1 |)$

Step 6.2

Simplify.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Simplify $(15 + x) (5 | 2 y - 1 |)$ .

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Step 6.2.1.1.1

Apply the distributive property.

$15 (5 | 2 y - 1 |) + x (5 | 2 y - 1 |) = - \frac{4 x}{5 | 2 y - 1 |} (5 | 2 y - 1 |)$

Step 6.2.1.1.2

Simplify the expression.

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Step 6.2.1.1.2.1

Multiply $5$ by $15$ .

$75 | 2 y - 1 | + x (5 | 2 y - 1 |) = - \frac{4 x}{5 | 2 y - 1 |} (5 | 2 y - 1 |)$

Step 6.2.1.1.2.2

Rewrite using the commutative property of multiplication.

$75 | 2 y - 1 | + 5 x | 2 y - 1 | = - \frac{4 x}{5 | 2 y - 1 |} (5 | 2 y - 1 |)$

Step 6.2.1.1.2.3

Reorder $75 | 2 y - 1 |$ and $5 x | 2 y - 1 |$ .

$5 x | 2 y - 1 | + 75 | 2 y - 1 | = - \frac{4 x}{5 | 2 y - 1 |} (5 | 2 y - 1 |)$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Cancel the common factor of $5 | 2 y - 1 |$ .

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Step 6.2.2.1.1

Move the leading negative in $- \frac{4 x}{5 | 2 y - 1 |}$ into the numerator.

$5 x | 2 y - 1 | + 75 | 2 y - 1 | = \frac{- 4 x}{5 | 2 y - 1 |} (5 | 2 y - 1 |)$

Step 6.2.2.1.2

Cancel the common factor.

$5 x | 2 y - 1 | + 75 | 2 y - 1 | = \frac{- 4 x}{5 | 2 y - 1 |} (5 | 2 y - 1 |)$

Step 6.2.2.1.3

Rewrite the expression.

$5 x | 2 y - 1 | + 75 | 2 y - 1 | = - 4 x$

Step 6.3

Solve for $x$ .

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Step 6.3.1

Add $4 x$ to both sides of the equation.

$5 x | 2 y - 1 | + 75 | 2 y - 1 | + 4 x = 0$

Step 6.3.2

Subtract $75 | 2 y - 1 |$ from both sides of the equation.

$5 x | 2 y - 1 | + 4 x = - 75 | 2 y - 1 |$

Step 6.3.3

Factor $x$ out of $5 x | 2 y - 1 | + 4 x$ .

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Step 6.3.3.1

Factor $x$ out of $5 x | 2 y - 1 |$ .

$x (5 | 2 y - 1 |) + 4 x = - 75 | 2 y - 1 |$

Step 6.3.3.2

Factor $x$ out of $4 x$ .

$x (5 | 2 y - 1 |) + x \cdot 4 = - 75 | 2 y - 1 |$

Step 6.3.3.3

Factor $x$ out of $x (5 | 2 y - 1 |) + x \cdot 4$ .

$x (5 | 2 y - 1 | + 4) = - 75 | 2 y - 1 |$

Step 6.3.4

Divide each term in $x (5 | 2 y - 1 | + 4) = - 75 | 2 y - 1 |$ by $5 | 2 y - 1 | + 4$ and simplify.

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Step 6.3.4.1

Divide each term in $x (5 | 2 y - 1 | + 4) = - 75 | 2 y - 1 |$ by $5 | 2 y - 1 | + 4$ .

$\frac{x (5 | 2 y - 1 | + 4)}{5 | 2 y - 1 | + 4} = \frac{- 75 | 2 y - 1 |}{5 | 2 y - 1 | + 4}$

Step 6.3.4.2

Simplify the left side.

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Step 6.3.4.2.1

Cancel the common factor of $5 | 2 y - 1 | + 4$ .

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Step 6.3.4.2.1.1

Cancel the common factor.

$\frac{x (5 | 2 y - 1 | + 4)}{5 | 2 y - 1 | + 4} = \frac{- 75 | 2 y - 1 |}{5 | 2 y - 1 | + 4}$

Step 6.3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 75 | 2 y - 1 |}{5 | 2 y - 1 | + 4}$

Step 6.3.4.3

Simplify the right side.

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Step 6.3.4.3.1

Move the negative in front of the fraction.

$x = - \frac{75 | 2 y - 1 |}{5 | 2 y - 1 | + 4}$

Step 7

List all of the solutions.

$x = - \frac{75 | 2 y - 1 |}{5 | 2 y - 1 | - 4}$

$x = - \frac{75 | 2 y - 1 |}{5 | 2 y - 1 | + 4}$