Precalculus Examples

$| x + 4 | = | 2 x + 1 |$

Step 1

Rewrite the equation as $| 2 x + 1 | = | x + 4 |$ .

$| 2 x + 1 | = | x + 4 |$

Step 2

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

$2 x + 1 = x + 4$

$2 x + 1 = - (x + 4)$

$- (2 x + 1) = x + 4$

$- (2 x + 1) = - (x + 4)$

Step 3

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

$2 x + 1 = x + 4$

$2 x + 1 = - (x + 4)$

Step 4

Solve $2 x + 1 = x + 4$ 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 $x$ from both sides of the equation.

$2 x + 1 - x = 4$

Step 4.1.2

Subtract $x$ from $2 x$ .

$x + 1 = 4$

Step 4.2

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

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$x = 4 - 1$

Step 4.2.2

Subtract $1$ from $4$ .

$x = 3$

Step 5

Solve $2 x + 1 = - (x + 4)$ for $x$ .

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Step 5.1

Simplify $- (x + 4)$ .

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Step 5.1.1

Rewrite.

$2 x + 1 = 0 + 0 - (x + 4)$

Step 5.1.2

Simplify by adding zeros.

$2 x + 1 = - (x + 4)$

Step 5.1.3

Apply the distributive property.

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

Step 5.1.4

Multiply $- 1$ by $4$ .

$2 x + 1 = - x - 4$

Step 5.2

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

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Step 5.2.1

Add $x$ to both sides of the equation.

$2 x + 1 + x = - 4$

Step 5.2.2

Add $2 x$ and $x$ .

$3 x + 1 = - 4$

Step 5.3

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

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Step 5.3.1

Subtract $1$ from both sides of the equation.

$3 x = - 4 - 1$

Step 5.3.2

Subtract $1$ from $- 4$ .

$3 x = - 5$

Step 5.4

Divide each term in $3 x = - 5$ by $3$ and simplify.

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Step 5.4.1

Divide each term in $3 x = - 5$ by $3$ .

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $3$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide $x$ by $1$ .

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

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.

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

Step 6

List all of the solutions.

$x = 3, - \frac{5}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 3, - \frac{5}{3}$

Decimal Form:

$x = 3, - 1.\overline{6}$

Mixed Number Form:

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