Trigonometry Examples

$| 2 x - 3 | = x + 1$

Step 1

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

$2 x - 3 = \pm (x + 1)$

Step 2

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

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Step 2.1

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

$2 x - 3 = x + 1$

Step 2.2

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

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Step 2.2.1

Subtract $x$ from both sides of the equation.

$2 x - 3 - x = 1$

Step 2.2.2

Subtract $x$ from $2 x$ .

$x - 3 = 1$

Step 2.3

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

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Step 2.3.1

Add $3$ to both sides of the equation.

$x = 1 + 3$

Step 2.3.2

Add $1$ and $3$ .

$x = 4$

Step 2.4

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

$2 x - 3 = - (x + 1)$

Step 2.5

Simplify $- (x + 1)$ .

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Step 2.5.1

Rewrite.

$2 x - 3 = 0 + 0 - (x + 1)$

Step 2.5.2

Simplify by adding zeros.

$2 x - 3 = - (x + 1)$

Step 2.5.3

Apply the distributive property.

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

Step 2.5.4

Multiply $- 1$ by $1$ .

$2 x - 3 = - x - 1$

Step 2.6

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

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Step 2.6.1

Add $x$ to both sides of the equation.

$2 x - 3 + x = - 1$

Step 2.6.2

Add $2 x$ and $x$ .

$3 x - 3 = - 1$

Step 2.7

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

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Step 2.7.1

Add $3$ to both sides of the equation.

$3 x = - 1 + 3$

Step 2.7.2

Add $- 1$ and $3$ .

$3 x = 2$

Step 2.8

Divide each term in $3 x = 2$ by $3$ and simplify.

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Step 2.8.1

Divide each term in $3 x = 2$ by $3$ .

$\frac{3 x}{3} = \frac{2}{3}$

Step 2.8.2

Simplify the left side.

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Step 2.8.2.1

Cancel the common factor of $3$ .

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Step 2.8.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2}{3}$

Step 2.8.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{3}$

Step 2.9

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

$x = 4, \frac{2}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 4, \frac{2}{3}$

Decimal Form:

$x = 4, 0.\overline{6}$