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Solution - Absolute value equations

Exact form:

x = 0

x=0

See steps

Step-by-step explanation

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

$- | 2 x + 5 | + | 2 x - 5 | = 0$

Add $- | 2 x - 5 |$ to both sides of the equation:

$- | 2 x + 5 | + | 2 x - 5 | - | 2 x - 5 | = - | 2 x - 5 |$

Simplify the arithmetic

$- | 2 x + 5 | = - | 2 x - 5 |$

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
$- | 2 x + 5 | = - | 2 x - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$- \| 2 x + 5 \| = - \| 2 x - 5 \|$
$x = + y$	$- (2 x + 5) = - (2 x - 5)$
$x = - y$	$- (2 x + 5) = - - (2 x - 5)$
$+ x = y$	$- (2 x + 5) = - (2 x - 5)$
$- x = y$	$- (- (2 x + 5)) = - (2 x - 5)$

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:

$\| x \| = \| y \|$	$- \| 2 x + 5 \| = - \| 2 x - 5 \|$
$x = + y$ , $+ x = y$	$- (2 x + 5) = - (2 x - 5)$
$x = - y$ , $- x = y$	$- (2 x + 5) = - - (2 x - 5)$

3. Solve the two equations for x

7 additional steps

$- (2 x + 5) = - (2 x - 5)$

Expand the parentheses:

$- 2 x - 5 = - (2 x - 5)$

Expand the parentheses:

$- 2 x - 5 = - 2 x + 5$

Add to both sides:

$(- 2 x - 5) + 2 x = (- 2 x + 5) + 2 x$

Group like terms:

$(- 2 x + 2 x) - 5 = (- 2 x + 5) + 2 x$

Simplify the arithmetic:

$- 5 = (- 2 x + 5) + 2 x$

Group like terms:

$- 5 = (- 2 x + 2 x) + 5$

Simplify the arithmetic:

$- 5 = 5$

The statement is false:

$- 5 = 5$

The equation is false so it has no solution.

10 additional steps

$- (2 x + 5) = - (- (2 x - 5))$

Expand the parentheses:

$- 2 x - 5 = - (- (2 x - 5))$

Resolve the double minus:

$- 2 x - 5 = 2 x - 5$

Subtract from both sides:

$(- 2 x - 5) - 2 x = (2 x - 5) - 2 x$

Group like terms:

$(- 2 x - 2 x) - 5 = (2 x - 5) - 2 x$

Simplify the arithmetic:

$- 4 x - 5 = (2 x - 5) - 2 x$

Group like terms:

$- 4 x - 5 = (2 x - 2 x) - 5$

Simplify the arithmetic:

$- 4 x - 5 = - 5$

Add to both sides:

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

Simplify the arithmetic:

$- 4 x = - 5 + 5$

Simplify the arithmetic:

$- 4 x = 0$

Divide both sides by the coefficient:

$x = 0$

4. Graph

Each line represents the function of one side of the equation:
$y = - | 2 x + 5 |$
$y = - | 2 x - 5 |$
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. 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. Graph

Why learn this

Terms and topics

Related links

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