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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

x = - 2, 0.667

x=-2 , 0.667

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| x - 2 | - 2 | x | = 0$

Add $2 | x |$ to both sides of the equation:

$| x - 2 | - 2 | x | + 2 | x | = 2 | x |$

Simplify the arithmetic

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

2. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| x - 2 | = 2 | x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x - 2 \| = 2 \| x \|$
$x = + y$	$(x - 2) = 2 (x)$
$x = - y$	$(x - 2) = 2 (- (x))$
$+ x = y$	$(x - 2) = 2 (x)$
$- x = y$	$- (x - 2) = 2 (x)$

When simplified, equations $x = + y$ and $+ x = y$ are the same and equations $x = - y$ and $- x = y$ are the same, so we end up with only 2 equations:

$\| x \| = \| y \|$	$\| x - 2 \| = 2 \| x \|$
$x = + y$ , $+ x = y$	$(x - 2) = 2 (x)$
$x = - y$ , $- x = y$	$(x - 2) = 2 (- (x))$

3. Solve the two equations for x

9 additional steps

$(x - 2) = 2 x$

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

$- x - 2 = 0$

Add to both sides:

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

Simplify the arithmetic:

$- x = 0 + 2$

Simplify the arithmetic:

$- x = 2$

Multiply both sides by :

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

Remove the one(s):

$x = 2 \cdot - 1$

Simplify the arithmetic:

$x = - 2$

10 additional steps

$(x - 2) = 2 \cdot - x$

Group like terms:

$(x - 2) = (2 \cdot - 1) x$

Multiply the coefficients:

$(x - 2) = - 2 x$

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

$3 x - 2 = 0$

Add to both sides:

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

Simplify the arithmetic:

$3 x = 0 + 2$

Simplify the arithmetic:

$3 x = 2$

Divide both sides by :

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

Simplify the fraction:

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

4. List the solutions

$x = - 2, \frac{2}{3}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | x - 2 |$
$y = 2 | x |$
The equation is true where the two lines cross.

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Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved