Precalculus Examples

$| 3 x + 4 | = | 2 x - 3 |$

Step 1

Rewrite the equation as $| 2 x - 3 | = | 3 x + 4 |$ .

$| 2 x - 3 | = | 3 x + 4 |$

Step 2

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

$2 x - 3 = 3 x + 4$

$2 x - 3 = - (3 x + 4)$

$- (2 x - 3) = 3 x + 4$

$- (2 x - 3) = - (3 x + 4)$

Step 3

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

$2 x - 3 = 3 x + 4$

$2 x - 3 = - (3 x + 4)$

Step 4

Solve $2 x - 3 = 3 x + 4$ for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $3 x$ from both sides of the equation.

$2 x - 3 - 3 x = 4$

Step 4.1.2

Subtract $3 x$ from $2 x$ .

$- x - 3 = 4$

Step 4.2

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

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Step 4.2.1

Add $3$ to both sides of the equation.

$- x = 4 + 3$

Step 4.2.2

Add $4$ and $3$ .

$- x = 7$

Step 4.3

Divide each term in $- x = 7$ by $- 1$ and simplify.

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Step 4.3.1

Divide each term in $- x = 7$ by $- 1$ .

$\frac{- x}{- 1} = \frac{7}{- 1}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{7}{- 1}$

Step 4.3.2.2

Divide $x$ by $1$ .

$x = \frac{7}{- 1}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $7$ by $- 1$ .

$x = - 7$

Step 5

Solve $2 x - 3 = - (3 x + 4)$ for $x$ .

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Step 5.1

Simplify $- (3 x + 4)$ .

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Step 5.1.1

Rewrite.

$2 x - 3 = 0 + 0 - (3 x + 4)$

Step 5.1.2

Simplify by adding zeros.

$2 x - 3 = - (3 x + 4)$

Step 5.1.3

Apply the distributive property.

$2 x - 3 = - (3 x) - 1 \cdot 4$

Step 5.1.4

Multiply.

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Step 5.1.4.1

Multiply $3$ by $- 1$ .

$2 x - 3 = - 3 x - 1 \cdot 4$

Step 5.1.4.2

Multiply $- 1$ by $4$ .

$2 x - 3 = - 3 x - 4$

Step 5.2

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

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Step 5.2.1

Add $3 x$ to both sides of the equation.

$2 x - 3 + 3 x = - 4$

Step 5.2.2

Add $2 x$ and $3 x$ .

$5 x - 3 = - 4$

Step 5.3

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

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Step 5.3.1

Add $3$ to both sides of the equation.

$5 x = - 4 + 3$

Step 5.3.2

Add $- 4$ and $3$ .

$5 x = - 1$

Step 5.4

Divide each term in $5 x = - 1$ by $5$ and simplify.

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Step 5.4.1

Divide each term in $5 x = - 1$ by $5$ .

$\frac{5 x}{5} = \frac{- 1}{5}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $5$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 1}{5}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{5}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{5}$

Step 6

List all of the solutions.

$x = - 7, - \frac{1}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - 7, - \frac{1}{5}$

Decimal Form:

$x = - 7, - 0.2$