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Solution - Absolute value equations

Exact form:

x = 2, 2

x=2 , 2

See steps

Step-by-step explanation

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

$| 4 x - 8 | - | - 3 x + 6 | = 0$

Add $| - 3 x + 6 |$ to both sides of the equation:

$| 4 x - 8 | - | - 3 x + 6 | + | - 3 x + 6 | = | - 3 x + 6 |$

Simplify the arithmetic

$| 4 x - 8 | = | - 3 x + 6 |$

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
$| 4 x - 8 | = | - 3 x + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - 8 \| = \| - 3 x + 6 \|$
$x = + y$	$(4 x - 8) = (- 3 x + 6)$
$x = - y$	$(4 x - 8) = (- (- 3 x + 6))$
$+ x = y$	$(4 x - 8) = (- 3 x + 6)$
$- x = y$	$- (4 x - 8) = (- 3 x + 6)$

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:

$\| x \| = \| y \|$	$\| 4 x - 8 \| = \| - 3 x + 6 \|$
$x = + y$ , $+ x = y$	$(4 x - 8) = (- 3 x + 6)$
$x = - y$ , $- x = y$	$(4 x - 8) = (- (- 3 x + 6))$

3. Solve the two equations for x

11 additional steps

$(4 x - 8) = (- 3 x + 6)$

Add to both sides:

$(4 x - 8) + 3 x = (- 3 x + 6) + 3 x$

Group like terms:

$(4 x + 3 x) - 8 = (- 3 x + 6) + 3 x$

Simplify the arithmetic:

$7 x - 8 = (- 3 x + 6) + 3 x$

Group like terms:

$7 x - 8 = (- 3 x + 3 x) + 6$

Simplify the arithmetic:

$7 x - 8 = 6$

Add to both sides:

$(7 x - 8) + 8 = 6 + 8$

Simplify the arithmetic:

$7 x = 6 + 8$

Simplify the arithmetic:

$7 x = 14$

Divide both sides by :

$\frac{(7 x)}{7} = \frac{14}{7}$

Simplify the fraction:

$x = \frac{14}{7}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(2 \cdot 7)}{(1 \cdot 7)}$

Factor out and cancel the greatest common factor:

$x = 2$

8 additional steps

$(4 x - 8) = - (- 3 x + 6)$

Expand the parentheses:

$(4 x - 8) = 3 x - 6$

Subtract from both sides:

$(4 x - 8) - 3 x = (3 x - 6) - 3 x$

Group like terms:

$(4 x - 3 x) - 8 = (3 x - 6) - 3 x$

Simplify the arithmetic:

$x - 8 = (3 x - 6) - 3 x$

Group like terms:

$x - 8 = (3 x - 3 x) - 6$

Simplify the arithmetic:

$x - 8 = - 6$

Add to both sides:

$(x - 8) + 8 = - 6 + 8$

Simplify the arithmetic:

$x = - 6 + 8$

Simplify the arithmetic:

$x = 2$

4. List the solutions

$x = 2, 2$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - 8 |$
$y = | - 3 x + 6 |$
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

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