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Solution - Absolute value equations

Exact form:

x = 1

x=1

See steps

Step-by-step explanation

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

$| - x + 4 | - | x + 2 | = 0$

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

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

Simplify the arithmetic

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

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

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

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:

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

3. Solve the two equations for x

12 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 2 x + 4 = 2$

Subtract from both sides:

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

Simplify the arithmetic:

$- 2 x = 2 - 4$

Simplify the arithmetic:

$- 2 x = - 2$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

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

Simplify the fraction:

$x = 1$

6 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$4 = - 2$

The statement is false:

$4 = - 2$

The equation is false so it has no solution.

4. List the solutions

$x = 1$
(1 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | - x + 4 |$
$y = | x + 2 |$
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 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