Algebra Examples

$| x + 7 | = | x + 11 |$

Step 1

Rewrite the equation as $| x + 11 | = | x + 7 |$ .

$| x + 11 | = | x + 7 |$

Step 2

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

$x + 11 = x + 7$

$x + 11 = - (x + 7)$

$- (x + 11) = x + 7$

$- (x + 11) = - (x + 7)$

Step 3

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

$x + 11 = x + 7$

$x + 11 = - (x + 7)$

Step 4

Solve $x + 11 = x + 7$ 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.

$x + 11 - x = 7$

Step 4.1.2

Combine the opposite terms in $x + 11 - x$ .

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Step 4.1.2.1

Subtract $x$ from $x$ .

$0 + 11 = 7$

Step 4.1.2.2

Add $0$ and $11$ .

$11 = 7$

Step 4.2

Since $11 \neq 7$ , there are no solutions.

No solution

Step 5

Solve $x + 11 = - (x + 7)$ for $x$ .

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Step 5.1

Simplify $- (x + 7)$ .

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Step 5.1.1

Rewrite.

$x + 11 = 0 + 0 - (x + 7)$

Step 5.1.2

Simplify by adding zeros.

$x + 11 = - (x + 7)$

Step 5.1.3

Apply the distributive property.

$x + 11 = - x - 1 \cdot 7$

Step 5.1.4

Multiply $- 1$ by $7$ .

$x + 11 = - x - 7$

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.

$x + 11 + x = - 7$

Step 5.2.2

Add $x$ and $x$ .

$2 x + 11 = - 7$

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 $11$ from both sides of the equation.

$2 x = - 7 - 11$

Step 5.3.2

Subtract $11$ from $- 7$ .

$2 x = - 18$

Step 5.4

Divide each term in $2 x = - 18$ by $2$ and simplify.

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Step 5.4.1

Divide each term in $2 x = - 18$ by $2$ .

$\frac{2 x}{2} = \frac{- 18}{2}$

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.

$\frac{2 x}{2} = \frac{- 18}{2}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 18}{2}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Divide $- 18$ by $2$ .

$x = - 9$

Step 6

List all of the solutions.

$x = - 9$

Step 7

Verify each of the solutions by substituting them into $| x + 7 | = | x + 11 |$ and solving.

$x = - 9$