Pre-Algebra Examples

${(x - 4)}^{2} = {(x - 2)}^{2}$

Step 1

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| x - 4 | = | x - 2 |$

Step 2

Solve for $x$ .

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Step 2.1

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

$x - 4 = x - 2$

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

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

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

Step 2.2

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

$x - 4 = x - 2$

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

Step 2.3

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

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Step 2.3.1

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

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Step 2.3.1.1

Subtract $x$ from both sides of the equation.

$x - 4 - x = - 2$

Step 2.3.1.2

Combine the opposite terms in $x - 4 - x$ .

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Step 2.3.1.2.1

Subtract $x$ from $x$ .

$0 - 4 = - 2$

Step 2.3.1.2.2

Subtract $4$ from $0$ .

$- 4 = - 2$

Step 2.3.2

Since $- 4 \neq - 2$ , there are no solutions.

No solution

Step 2.4

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

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Step 2.4.1

Simplify $- (x - 2)$ .

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Step 2.4.1.1

Rewrite.

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

Step 2.4.1.2

Simplify by adding zeros.

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

Step 2.4.1.3

Apply the distributive property.

$x - 4 = - x - - 2$

Step 2.4.1.4

Multiply $- 1$ by $- 2$ .

$x - 4 = - x + 2$

Step 2.4.2

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

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Step 2.4.2.1

Add $x$ to both sides of the equation.

$x - 4 + x = 2$

Step 2.4.2.2

Add $x$ and $x$ .

$2 x - 4 = 2$

Step 2.4.3

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

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Step 2.4.3.1

Add $4$ to both sides of the equation.

$2 x = 2 + 4$

Step 2.4.3.2

Add $2$ and $4$ .

$2 x = 6$

Step 2.4.4

Divide each term in $2 x = 6$ by $2$ and simplify.

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Step 2.4.4.1

Divide each term in $2 x = 6$ by $2$ .

$\frac{2 x}{2} = \frac{6}{2}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor of $2$ .

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Step 2.4.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{6}{2}$

Step 2.4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{2}$

Step 2.4.4.3

Simplify the right side.

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Step 2.4.4.3.1

Divide $6$ by $2$ .

$x = 3$

Step 2.5

List all of the solutions.

$x = 3$