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Solution - Absolute value equations

Exact form:

x = \frac{8}{5}, 4

x=\frac{8}{5} , 4

Mixed number form:

x = 1 \frac{3}{5}, 4

x=1\frac{3}{5} , 4

Decimal form:

x = 1.6, 4

x=1.6 , 4

See steps

Step-by-step explanation

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

$2 | x - 1 | - 3 | - x + 2 | = 0$

Add $3 | - x + 2 |$ to both sides of the equation:

$2 | x - 1 | - 3 | - x + 2 | + 3 | - x + 2 | = 3 | - x + 2 |$

Simplify the arithmetic

$2 | x - 1 | = 3 | - x + 2 |$

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
$2 | x - 1 | = 3 | - x + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$2 \| x - 1 \| = 3 \| - x + 2 \|$
$x = + y$	$2 (x - 1) = 3 (- x + 2)$
$x = - y$	$2 (x - 1) = 3 (- (- x + 2))$
$+ x = y$	$2 (x - 1) = 3 (- x + 2)$
$- x = y$	$2 (- (x - 1)) = 3 (- x + 2)$

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:

$\| x \| = \| y \|$	$2 \| x - 1 \| = 3 \| - x + 2 \|$
$x = + y$ , $+ x = y$	$2 (x - 1) = 3 (- x + 2)$
$x = - y$ , $- x = y$	$2 (x - 1) = 3 (- (- x + 2))$

3. Solve the two equations for x

15 additional steps

$2 \cdot (x - 1) = 3 \cdot (- x + 2)$

Expand the parentheses:

$2 x + 2 \cdot - 1 = 3 \cdot (- x + 2)$

Simplify the arithmetic:

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

Expand the parentheses:

$2 x - 2 = 3 \cdot - x + 3 \cdot 2$

Group like terms:

$2 x - 2 = (3 \cdot - 1) x + 3 \cdot 2$

Multiply the coefficients:

$2 x - 2 = - 3 x + 3 \cdot 2$

Simplify the arithmetic:

$2 x - 2 = - 3 x + 6$

Add to both sides:

$(2 x - 2) + 3 x = (- 3 x + 6) + 3 x$

Group like terms:

$(2 x + 3 x) - 2 = (- 3 x + 6) + 3 x$

Simplify the arithmetic:

$5 x - 2 = (- 3 x + 6) + 3 x$

Group like terms:

$5 x - 2 = (- 3 x + 3 x) + 6$

Simplify the arithmetic:

$5 x - 2 = 6$

Add to both sides:

$(5 x - 2) + 2 = 6 + 2$

Simplify the arithmetic:

$5 x = 6 + 2$

Simplify the arithmetic:

$5 x = 8$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{8}{5}$

Simplify the fraction:

$x = \frac{8}{5}$

15 additional steps

$2 \cdot (x - 1) = 3 \cdot (- (- x + 2))$

Expand the parentheses:

$2 x + 2 \cdot - 1 = 3 \cdot (- (- x + 2))$

Simplify the arithmetic:

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

Expand the parentheses:

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

$2 x - 2 = 3 x + 3 \cdot - 2$

Simplify the arithmetic:

$2 x - 2 = 3 x - 6$

Subtract from both sides:

$(2 x - 2) - 3 x = (3 x - 6) - 3 x$

Group like terms:

$(2 x - 3 x) - 2 = (3 x - 6) - 3 x$

Simplify the arithmetic:

$- x - 2 = (3 x - 6) - 3 x$

Group like terms:

$- x - 2 = (3 x - 3 x) - 6$

Simplify the arithmetic:

$- x - 2 = - 6$

Add to both sides:

$(- x - 2) + 2 = - 6 + 2$

Simplify the arithmetic:

$- x = - 6 + 2$

Simplify the arithmetic:

$- x = - 4$

Multiply both sides by :

$- x \cdot - 1 = - 4 \cdot - 1$

Remove the one(s):

$x = - 4 \cdot - 1$

Simplify the arithmetic:

$x = 4$

4. List the solutions

$x = \frac{8}{5}, 4$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = 2 | x - 1 |$
$y = 3 | - x + 2 |$
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