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Solution - Absolute value equations

Exact form:

x = 0

x=0

See steps

Step-by-step explanation

1. 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
$2 | 2 x - 1 | = | 4 x + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$2 \| 2 x - 1 \| = \| 4 x + 2 \|$
$x = + y$	$2 (2 x - 1) = (4 x + 2)$
$x = - y$	$2 (2 x - 1) = - (4 x + 2)$
$+ x = y$	$2 (2 x - 1) = (4 x + 2)$
$- x = y$	$2 (- (2 x - 1)) = (4 x + 2)$

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:

$\| x \| = \| y \|$	$2 \| 2 x - 1 \| = \| 4 x + 2 \|$
$x = + y$ , $+ x = y$	$2 (2 x - 1) = (4 x + 2)$
$x = - y$ , $- x = y$	$2 (2 x - 1) = - (4 x + 2)$

2. Solve the two equations for x

8 additional steps

$2 \cdot (2 x - 1) = (4 x + 2)$

Expand the parentheses:

$2 \cdot 2 x + 2 \cdot - 1 = (4 x + 2)$

Multiply the coefficients:

$4 x + 2 \cdot - 1 = (4 x + 2)$

Simplify the arithmetic:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 2 = 2$

The statement is false:

$- 2 = 2$

The equation is false so it has no solution.

12 additional steps

$2 \cdot (2 x - 1) = - (4 x + 2)$

Expand the parentheses:

$2 \cdot 2 x + 2 \cdot - 1 = - (4 x + 2)$

Multiply the coefficients:

$4 x + 2 \cdot - 1 = - (4 x + 2)$

Simplify the arithmetic:

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$8 x - 2 = (- 4 x - 2) + 4 x$

Group like terms:

$8 x - 2 = (- 4 x + 4 x) - 2$

Simplify the arithmetic:

$8 x - 2 = - 2$

Add to both sides:

$(8 x - 2) + 2 = - 2 + 2$

Simplify the arithmetic:

$8 x = - 2 + 2$

Simplify the arithmetic:

$8 x = 0$

Divide both sides by the coefficient:

$x = 0$

3. Graph

Each line represents the function of one side of the equation:
$y = 2 | 2 x - 1 |$
$y = | 4 x + 2 |$
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 without absolute value bars

2. Solve the two equations for x

3. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. Graph

Why learn this

Terms and topics

Related links

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