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Solution - Absolute value equations

Exact form:

x = - 7, 5

x=-7 , 5

See steps

Step-by-step explanation

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

$3 | x - 3 | - | 2 x - 16 | = 0$

Add $| 2 x - 16 |$ to both sides of the equation:

$3 | x - 3 | - | 2 x - 16 | + | 2 x - 16 | = | 2 x - 16 |$

Simplify the arithmetic

$3 | x - 3 | = | 2 x - 16 |$

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
$3 | x - 3 | = | 2 x - 16 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| x - 3 \| = \| 2 x - 16 \|$
$x = + y$	$3 (x - 3) = (2 x - 16)$
$x = - y$	$3 (x - 3) = (- (2 x - 16))$
$+ x = y$	$3 (x - 3) = (2 x - 16)$
$- x = y$	$3 (- (x - 3)) = (2 x - 16)$

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:

$\| x \| = \| y \|$	$3 \| x - 3 \| = \| 2 x - 16 \|$
$x = + y$ , $+ x = y$	$3 (x - 3) = (2 x - 16)$
$x = - y$ , $- x = y$	$3 (x - 3) = (- (2 x - 16))$

3. Solve the two equations for x

9 additional steps

$3 \cdot (x - 3) = (2 x - 16)$

Expand the parentheses:

$3 x + 3 \cdot - 3 = (2 x - 16)$

Simplify the arithmetic:

$3 x - 9 = (2 x - 16)$

Subtract from both sides:

$(3 x - 9) - 2 x = (2 x - 16) - 2 x$

Group like terms:

$(3 x - 2 x) - 9 = (2 x - 16) - 2 x$

Simplify the arithmetic:

$x - 9 = (2 x - 16) - 2 x$

Group like terms:

$x - 9 = (2 x - 2 x) - 16$

Simplify the arithmetic:

$x - 9 = - 16$

Add to both sides:

$(x - 9) + 9 = - 16 + 9$

Simplify the arithmetic:

$x = - 16 + 9$

Simplify the arithmetic:

$x = - 7$

14 additional steps

$3 \cdot (x - 3) = (- (2 x - 16))$

Expand the parentheses:

$3 x + 3 \cdot - 3 = (- (2 x - 16))$

Simplify the arithmetic:

$3 x - 9 = (- (2 x - 16))$

Expand the parentheses:

$3 x - 9 = - 2 x + 16$

Add to both sides:

$(3 x - 9) + 2 x = (- 2 x + 16) + 2 x$

Group like terms:

$(3 x + 2 x) - 9 = (- 2 x + 16) + 2 x$

Simplify the arithmetic:

$5 x - 9 = (- 2 x + 16) + 2 x$

Group like terms:

$5 x - 9 = (- 2 x + 2 x) + 16$

Simplify the arithmetic:

$5 x - 9 = 16$

Add to both sides:

$(5 x - 9) + 9 = 16 + 9$

Simplify the arithmetic:

$5 x = 16 + 9$

Simplify the arithmetic:

$5 x = 25$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{25}{5}$

Simplify the fraction:

$x = \frac{25}{5}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = 5$

4. List the solutions

$x = - 7, 5$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = 3 | x - 3 |$
$y = | 2 x - 16 |$
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