Precalculus Examples

$| x + 4 | = | x - 5 |$

Step 1

Rewrite the equation as $| x - 5 | = | x + 4 |$ .

$| x - 5 | = | x + 4 |$

Step 2

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

$x - 5 = x + 4$

$x - 5 = - (x + 4)$

$- (x - 5) = x + 4$

$- (x - 5) = - (x + 4)$

Step 3

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

$x - 5 = x + 4$

$x - 5 = - (x + 4)$

Step 4

Solve $x - 5 = x + 4$ 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.

$x - 5 - x = 4$

Step 4.1.2

Combine the opposite terms in $x - 5 - x$ .

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Step 4.1.2.1

Subtract $x$ from $x$ .

$0 - 5 = 4$

Step 4.1.2.2

Subtract $5$ from $0$ .

$- 5 = 4$

Step 4.2

Since $- 5 \neq 4$ , there are no solutions.

No solution

Step 5

Solve $x - 5 = - (x + 4)$ for $x$ .

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Step 5.1

Simplify $- (x + 4)$ .

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Step 5.1.1

Rewrite.

$x - 5 = 0 + 0 - (x + 4)$

Step 5.1.2

Simplify by adding zeros.

$x - 5 = - (x + 4)$

Step 5.1.3

Apply the distributive property.

$x - 5 = - x - 1 \cdot 4$

Step 5.1.4

Multiply $- 1$ by $4$ .

$x - 5 = - x - 4$

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.

$x - 5 + x = - 4$

Step 5.2.2

Add $x$ and $x$ .

$2 x - 5 = - 4$

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 $5$ to both sides of the equation.

$2 x = - 4 + 5$

Step 5.3.2

Add $- 4$ and $5$ .

$2 x = 1$

Step 5.4

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 5.4.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}$

Decimal Form:

$x = 0.5$