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Solution - Absolute value equations

Exact form:

x = 4, 4

x=4 , 4

See steps

Step-by-step explanation

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

$3 | x - 4 | + 2 | x - 4 | = 0$

Add $- 2 | x - 4 |$ to both sides of the equation:

$3 | x - 4 | + 2 | x - 4 | - 2 | x - 4 | = - 2 | x - 4 |$

Simplify the arithmetic

$3 | x - 4 | = - 2 | x - 4 |$

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
$3 | x - 4 | = - 2 | x - 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| x - 4 \| = - 2 \| x - 4 \|$
$x = + y$	$3 (x - 4) = - 2 (x - 4)$
$x = - y$	$3 (x - 4) = - 2 (- (x - 4))$
$+ x = y$	$3 (x - 4) = - 2 (x - 4)$
$- x = y$	$3 (- (x - 4)) = - 2 (x - 4)$

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:

$\| x \| = \| y \|$	$3 \| x - 4 \| = - 2 \| x - 4 \|$
$x = + y$ , $+ x = y$	$3 (x - 4) = - 2 (x - 4)$
$x = - y$ , $- x = y$	$3 (x - 4) = - 2 (- (x - 4))$

3. Solve the two equations for x

15 additional steps

$3 \cdot (x - 4) = - 2 \cdot (x - 4)$

Expand the parentheses:

$3 x + 3 \cdot - 4 = - 2 \cdot (x - 4)$

Simplify the arithmetic:

$3 x - 12 = - 2 \cdot (x - 4)$

Expand the parentheses:

$3 x - 12 = - 2 x - 2 \cdot - 4$

Simplify the arithmetic:

$3 x - 12 = - 2 x + 8$

Add to both sides:

$(3 x - 12) + 2 x = (- 2 x + 8) + 2 x$

Group like terms:

$(3 x + 2 x) - 12 = (- 2 x + 8) + 2 x$

Simplify the arithmetic:

$5 x - 12 = (- 2 x + 8) + 2 x$

Group like terms:

$5 x - 12 = (- 2 x + 2 x) + 8$

Simplify the arithmetic:

$5 x - 12 = 8$

Add to both sides:

$(5 x - 12) + 12 = 8 + 12$

Simplify the arithmetic:

$5 x = 8 + 12$

Simplify the arithmetic:

$5 x = 20$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{20}{5}$

Simplify the fraction:

$x = \frac{20}{5}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(4 \cdot 5)}{(1 \cdot 5)}$

Factor out and cancel the greatest common factor:

$x = 4$

14 additional steps

$3 \cdot (x - 4) = - 2 \cdot (- (x - 4))$

Expand the parentheses:

$3 x + 3 \cdot - 4 = - 2 \cdot (- (x - 4))$

Simplify the arithmetic:

$3 x - 12 = - 2 \cdot (- (x - 4))$

Expand the parentheses:

$3 x - 12 = - 2 \cdot (- x + 4)$

$3 x - 12 = - 2 \cdot - x - 2 \cdot 4$

Group like terms:

$3 x - 12 = (- 2 \cdot - 1) x - 2 \cdot 4$

Multiply the coefficients:

$3 x - 12 = 2 x - 2 \cdot 4$

Simplify the arithmetic:

$3 x - 12 = 2 x - 8$

Subtract from both sides:

$(3 x - 12) - 2 x = (2 x - 8) - 2 x$

Group like terms:

$(3 x - 2 x) - 12 = (2 x - 8) - 2 x$

Simplify the arithmetic:

$x - 12 = (2 x - 8) - 2 x$

Group like terms:

$x - 12 = (2 x - 2 x) - 8$

Simplify the arithmetic:

$x - 12 = - 8$

Add to both sides:

$(x - 12) + 12 = - 8 + 12$

Simplify the arithmetic:

$x = - 8 + 12$

Simplify the arithmetic:

$x = 4$

4. List the solutions

$x = 4, 4$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = 3 | x - 4 |$
$y = - 2 | x - 4 |$
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