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Solution - Absolute value equations

Exact form:

x = 8, 8

x=8 , 8

See steps

Step-by-step explanation

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

$| 3 x - 24 | - | - x + 8 | = 0$

Add $| - x + 8 |$ to both sides of the equation:

$| 3 x - 24 | - | - x + 8 | + | - x + 8 | = | - x + 8 |$

Simplify the arithmetic

$| 3 x - 24 | = | - x + 8 |$

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
$| 3 x - 24 | = | - x + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 x - 24 \| = \| - x + 8 \|$
$x = + y$	$(3 x - 24) = (- x + 8)$
$x = - y$	$(3 x - 24) = (- (- x + 8))$
$+ x = y$	$(3 x - 24) = (- x + 8)$
$- x = y$	$- (3 x - 24) = (- x + 8)$

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:

$\| x \| = \| y \|$	$\| 3 x - 24 \| = \| - x + 8 \|$
$x = + y$ , $+ x = y$	$(3 x - 24) = (- x + 8)$
$x = - y$ , $- x = y$	$(3 x - 24) = (- (- x + 8))$

3. Solve the two equations for x

11 additional steps

$(3 x - 24) = (- x + 8)$

Add to both sides:

$(3 x - 24) + x = (- x + 8) + x$

Group like terms:

$(3 x + x) - 24 = (- x + 8) + x$

Simplify the arithmetic:

$4 x - 24 = (- x + 8) + x$

Group like terms:

$4 x - 24 = (- x + x) + 8$

Simplify the arithmetic:

$4 x - 24 = 8$

Add to both sides:

$(4 x - 24) + 24 = 8 + 24$

Simplify the arithmetic:

$4 x = 8 + 24$

Simplify the arithmetic:

$4 x = 32$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{32}{4}$

Simplify the fraction:

$x = \frac{32}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(8 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = 8$

12 additional steps

$(3 x - 24) = - (- x + 8)$

Expand the parentheses:

$(3 x - 24) = x - 8$

Subtract from both sides:

$(3 x - 24) - x = (x - 8) - x$

Group like terms:

$(3 x - x) - 24 = (x - 8) - x$

Simplify the arithmetic:

$2 x - 24 = (x - 8) - x$

Group like terms:

$2 x - 24 = (x - x) - 8$

Simplify the arithmetic:

$2 x - 24 = - 8$

Add to both sides:

$(2 x - 24) + 24 = - 8 + 24$

Simplify the arithmetic:

$2 x = - 8 + 24$

Simplify the arithmetic:

$2 x = 16$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{16}{2}$

Simplify the fraction:

$x = \frac{16}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(8 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = 8$

4. List the solutions

$x = 8, 8$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 3 x - 24 |$
$y = | - x + 8 |$
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